\textbf{ĐỀ MINH HỌA LẦN 1} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\sqrt{2x-1}$ \choice {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{2}{3}(2x-1)\sqrt{2x-1}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{1}{3}(2x-1)\sqrt{2x-1}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=-\dfrac{1}{3}\sqrt{2x-1}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{1}{2}\sqrt{2x-1}+C}$} \end{ex} \textbf{ĐỀ MINH HỌA LẦN 2} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\cos 2x$ \choice {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x}=\dfrac{1}{2}\sin 2x+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=-\dfrac{1}{2}\sin 2x+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=2\sin 2x+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=-2\sin 2x+C$} \end{ex} \begin{ex} Biết $F(x)$ nguyên hàm $f(x)=\dfrac{1}{x-1}$ $F(2)=1$ Tính $F(3)$ \choice {$F(3)=\ln 2-1$} {\True $F(3)=\ln 2+1$} {$F(3)=\dfrac{1}{2}$} {$F(3)=\dfrac{7}{4}$} \end{ex} \textbf{ĐỀ MINH HỌA LẦN 3} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=x^2+\dfrac{2}{x^2}$ \choice {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{x^3}{3}-\dfrac{2}{x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{x^3}{3}-\dfrac{1}{x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{x^3}{3}+\dfrac{2}{x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{x^3}{3}+\dfrac{1}{x}+C}$} \end{ex} \textbf{ĐỀ THPT-QG 2017} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\dfrac{1}{5x-2}$ \choice {$\displaystyle\int\limits {\dfrac{1}{5x-2}\mathrm{\,d}x=\dfrac{1}{5}\ln |5x-2|+C}$} {$\displaystyle\int\limits {\dfrac{1}{5x-2}\mathrm{\,d}x=-\dfrac{1}{2}\ln (5x-2)+C}$} {$\displaystyle\int\limits {\dfrac{1}{5x-2}\mathrm{\,d}x=5\ln |5x-2|+C}$} {$\displaystyle\int\limits {\dfrac{1}{5x-2}\mathrm{\,d}x=\ln |5x-2|+C}$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=7^x$ \choice {$\displaystyle\int\limits {7^x \mathrm{\,d}x}=7^x\ln 7+C$} {$\displaystyle\int\limits {7^x \mathrm{\,d}x}=\dfrac{7^x}{\ln 7}+C$} {$\displaystyle\int\limits {7^x \mathrm{\,d}x}=7^{x+1}+C$} {$\displaystyle\int\limits {7^x \mathrm{\,d}x}=\dfrac{7^{x+1}}{x+1}+C$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=\cos 3x$ \choice {$\displaystyle\int\limits {\cos 3x \mathrm{\,d}x}=3\sin 3x+C$} {$\displaystyle\int\limits {\cos 3x \mathrm{\,d}x}=\dfrac{\sin 3x}{3}+C$} {$\displaystyle\int\limits {\cos 3x \mathrm{\,d}x}=\sin 3x+C$} {$\displaystyle\int\limits {\cos 3x \mathrm{\,d}x}=\cos 3x+C$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=2\sin x$ \choice {$\displaystyle\int\limits {2\sin x \mathrm{\,d}x}=2\cos x+C$} {$\displaystyle\int\limits {2\sin x \mathrm{\,d}x}=\sin^2x+C$} {$\displaystyle\int\limits {2\sin x \mathrm{\,d}x}=\sin 2x+C$} {$\displaystyle\int\limits {2\sin x \mathrm{\,d}x}=-2\cos x+C$} \end{ex} \begin{ex} Cho $F(x)=(x-1)\mathrm{e}^x$ nguyên hàm hàm số $f(x).\mathrm{e}^{2x}$ Tìm nguyên hàm hàm số $f'(x).\mathrm{e}^{2x}$ \choice {$\displaystyle\int\limits {f'(x).\mathrm{e}^{2x}}\mathrm{\,d}x=(4-2x)\mathrm{e}^x+C$} {$\displaystyle\int\limits {f'(x).\mathrm{e}^{2x}}\mathrm{\,d}x=\dfrac{2-x}{2}\mathrm{e}^x+C$} {$\displaystyle\int\limits {f'(x).\mathrm{e}^{2x}}\mathrm{\,d}x=(2-x)\mathrm{e}^x+C$} {$\displaystyle\int\limits {f'(x).\mathrm{e}^{2x}}\mathrm{\,d}x=(x-2)\mathrm{e}^x+C$} \end{ex} \begin{ex} Cho $F(x)=x^2$ nguyên hàm hàm số $f(x)\mathrm{e}^{2x}$ Tìm nguyên hàm hàm số $f'(x)\mathrm{e}^{2x}$ \choice {$\displaystyle\int\limits {{f'}}(x)\mathrm{e}^{2x}\mathrm{\,d}x=-x^2+2x+C$} {$\displaystyle\int\limits {{f'}}(x)\mathrm{e}^{2x}\mathrm{\,d}x=-x^2+x+C$} {$\displaystyle\int\limits {{f'}}(x)\mathrm{e}^{2x}\mathrm{\,d}x=2x^2-2x+C$} {$\displaystyle\int\limits {{f'}}(x)\mathrm{e}^{2x}\mathrm{\,d}x=-2x^2+2x+C$} \end{ex} \begin{ex} Cho $F(x)=\dfrac{1}{2x^2}$ nguyên hàm hàm số $\dfrac{f(x)}{x}$ Tìm nguyên hàm hàm số $f'(x)\ln x$ \choice {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x=-\left(\dfrac{\ln x}{x^2}+\dfrac{1}{2x^2}\right)}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x=\dfrac{\ln x}{x^2}+\dfrac{1}{x^2}}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x=-\left(\dfrac{\ln x}{x^2}+\dfrac{1}{x^2}\right)}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x=\dfrac{\ln x}{x^2}+\dfrac{1}{2x^2}}+C$} \end{ex} \begin{ex} Cho $F(x)=-\dfrac{1}{3x^2}$ nguyên hàm hàm số $\dfrac{f(x)}{x}$ Tìm nguyên hàm hàm số $f'(x)\ln x$ \choice {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x}=\dfrac{\ln x}{x^3}+\dfrac{1}{5x^5}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x}=\dfrac{\ln x}{x^3}-\dfrac{1}{5x^5}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x}=\dfrac{\ln x}{x^3}+\dfrac{1}{3x^3}+C$} {$\displaystyle\int\limits {f'(x)\ln x \mathrm{\,d}x}=-\dfrac{\ln x}{x^3}+\dfrac{1}{3x^3}+C$} \end{ex} \begin{ex} Tìm nguyên hàm $F(x)$ hàm số $f(x)=\sin x+\cos x$ thoả mãn $F\left(\dfrac{\pi}{2}\right)=2$ \choice {$F(x)=\cos x-\sin x+3$} {$F(x)=-\cos x+\sin x+3$} {$F(x)=-\cos x+\sin x-1$} {$F(x)=-\cos x+\sin x+1$} \end{ex} \begin{ex} Cho $F(x)$ nguyên hàm hàm số $f(x)=\mathrm{e}^x+2x$ thỏa mãn $F(0)=\dfrac{3}{2}$ Tìm $F(x)$ \choice {$F(x)=\mathrm{e}^x+x^2+\dfrac{3}{2}$} {$F(x)=2\mathrm{e}^x+x^2-\dfrac{1}{2}$} {$F(x)=\mathrm{e}^x+x^2+\dfrac{5}{2}$} {$F(x)=\mathrm{e}^x+x^2+\dfrac{1}{2}$} \end{ex} \begin{ex} Cho $F(x)$ nguyên hàm hàm số $f(x)=\dfrac{\ln x}{x}$ Tính $I=F(e)-F(1)$ \choice {$I=e$} {$I=\dfrac{1}{e}$} {$I=\dfrac{1}{2}$} {$I=1$} \end{ex} \begin{ex} Cho hàm số $y=f(x)$ thỏa mãn $f'(x)=3-5\sin x$ $f(0)=10$ Mệnh đề đúng? \choice {$f(x)=3x+5\cos x+5$} {$f(x)=3x+5\cos x+2$} {$f(x)=3x-5\cos x+2$} {$f(x)=3x-5\cos x+15$} \end{ex} \textbf{SỞ BÌNH THUẬN} \\ \begin{ex} Tìm ngun hàm hàm số $f(x)=\dfrac{1}{5x+1}$ \choice {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=\dfrac{1}{5}\ln (5x+1)+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=5\ln |5x+1|+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=\ln |5x+1|+C}$} {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x=\dfrac{1}{5}\ln |5x+1|+C}$} \end{ex} \begin{ex} Cho hàm số $f(x)=\cos x$ Tìm nguyên hàm hàm số $y={\left(f'(x)\right)}^2$ \choice {\True $\displaystyle\int\limits {y \mathrm{\,d}x=\dfrac{x}{2}-\dfrac{1}{4}\sin 2x+C}$} {$\displaystyle\int\limits {y \mathrm{\,d}x=\dfrac{x}{2}+\dfrac{1}{4}\sin 2x+C}$} {$\displaystyle\int\limits {y \mathrm{\,d}x=x+\dfrac{1}{2}\sin 2x+C}$} {$\displaystyle\int\limits {y \mathrm{\,d}x=x-\dfrac{1}{2}\sin 2x+C}$} \end{ex} \textbf{SỞ BÌNH PHƯỚC} \\ \begin{ex} Trong khẳng định sau, khẳng định \textbf{sai}? \choice {Nếu $f(x),\,g(x)$ hàm số liên tục R $\displaystyle\int\limits {\left[f(x) +g(x)\right]\mathrm{\,d}x=\displaystyle\int\limits {f(x)\mathrm{\,d}x+}}\displaystyle\int\limits {g(x)\mathrm{\,d}x}$} {Nếu $F(x)$ $G(x)$ nguyên hàm hàm số $f(x)$ $F(x)-G(x)=C$ (với C số)} {\True Nếu hàm số $u(x),\,v(x)$ liên tục có đạo hàm R $\displaystyle\int\limits {u(x)v'(x)\mathrm{\,d}x+\displaystyle\int\limits {v(x)u'(x)\mathrm{\,d}x=u(x)v(x)}}$} {$F(x)=x^2$ nguyên hàm $f(x)=2x$} \end{ex} \begin{ex} Tìm nguyên hàm $F(x)$ hàm số $f(x)=\cos 2x$, biết $F\left(\dfrac{\pi}{2}\right)=2\pi $ \choice {$F(x)=\sin x+2\pi $} {$F(x)=x+\sin 2x+\dfrac{3\pi}{2}$} {\True $F(x)=\dfrac{1}{2}\sin 2x+2\pi $} {$F(x)=2x+2\pi $} \end{ex} \textbf{SỞ THỪA THIÊN HUẾ} \\ \begin{ex} Tính nguyên hàm $I=\displaystyle\int\limits {\left(x^2+\dfrac{2}{x}-3\sqrt{x}\right) \mathrm{\,d}x}$ \choice {$I=\dfrac{x^3}{3}-2\ln |x|+2\sqrt{x^3}+C$} {$I=\dfrac{x^3}{3}+2\ln |x|+2\sqrt{x^3}+C$} {$I=\dfrac{x^3}{3}+2\ln x-2\sqrt{x^3}+C$} {\True $I=\dfrac{x^3}{3}+2\ln |x|-2\sqrt{x^3}+C$} \end{ex} \begin{ex} Cho $f(x)=\dfrac{x}{\sqrt{x^2+1}}(2\sqrt{x^2+1}+2017)$, biết F(x) nguyên hàm $f(x)$ thỏa mãn $F(0)=2018$ Tính $F(2)$ \choice {\True $F(2)=5+2017\sqrt{5}$} {$F(2)=4+2017\sqrt{4}$} {$F(2)=3+2017\sqrt{3}$} {$F(2)=2022$} \end{ex} \textbf{SỞ CAO BẰNG} \\ \begin{ex} Nguyên hàm hàm số $f(x)=\sin x\cos x$ là: \choice {$-\sin x\cos x$} {$-\dfrac{1}{4}\sin 2x+C$} {$\dfrac{1}{4}\cos 2x+C$} {\True $-\dfrac{1}{4}\cos 2x+C$} \end{ex} \begin{ex} Nguyên hàm $F(x)$ hàm số $f(x)=4x^3-3x^2+2$ thỏa mãn $F(-1)=3$ \choice {$x^4-x^3+2x$} {$x^4-x^3+2x-3$} {\True $x^4-x^3+2x+3$} {$x^4-x^3+2x+4$} \end{ex} \textbf{SỞ HƯNG YÊN LẦN 2} \\ \begin{ex} Tính \choice {$I=2\sin \left(2x+\dfrac{\pi}{3}\right)+C$} {$I=-\dfrac{1}{2}\sin \left(2x+\dfrac{\pi}{3}\right)+C$} {$I=-2\sin \left(2x+\dfrac{\pi}{3}\right)+C$} {$I=\dfrac{1}{2}\sin \left(2x+\dfrac{\pi}{3}\right)+C$} \end{ex} \begin{ex} Tìm hàm số $f(x)$ biết $f'(x)=\dfrac{2x+3}{x+1}$ $f(0)=1$ \choice {$f(x)=2x+\ln |2x+1|-1$} {$f(x)=x+\ln |x+1|+1$} {$f(x)=x^2+\ln |x+1|$} {$f(x)=2x+\ln |x+1|+1$} \end{ex} \textbf{SỞ ĐIỆN BIÊN} \\ \begin{ex} Tính nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ \choice {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{1}{2}\mathrm{e}^{2x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=2\mathrm{e}^{2x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=-2\mathrm{e}^{2x}+C}$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=-\dfrac{1}{2}\mathrm{e}^{2x}+C}$} \end{ex} \textbf{SỞ GIÁO DỤC ĐÀ NẴNG} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=tan^2x$ \choice {$F(x)=-\ln \left|\cos x\right|+C$} {$F(x)=x+\tan x+C$} {$F(x)=-x+\tan x+C$} {$F(x)=\ln \left|\cos x\right|+C$} \end{ex} \textbf{SỞ NAM ĐỊNH} \\ \begin{ex} Tìm hàm số $f(x)$ biết $f'(x)=\dfrac{\cos x}{{\left(2+\sin x\right)}^2}$ \choice {$f(x)=\dfrac{\sin x}{{\left(2+\sin x\right)}^2}+C$} {$f(x)=\dfrac{1}{2+\cos x}+C$} {\True $f(x)=-\dfrac{1}{2+\sin x}+C$} {$f(x)=\dfrac{\sin x}{2+\sin x}+C$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ \choice {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=2\mathrm{e}^{2x}+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=\dfrac{\mathrm{e}^{2x+1}}{2x+1}+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=2x\mathrm{e}^{2x-1}+C}$} {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x=\dfrac{\mathrm{e}^{2x}}{2}+C}$} \end{ex} \textbf{SỞ PHÚ THỌ} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ \choice {$\displaystyle\int\limits {\mathrm{e}^{2x}\mathrm{\,d}x}=-\dfrac{1}{2}\mathrm{e}^{2x}+C$} {\True $\displaystyle\int\limits {\mathrm{e}^{2x}\mathrm{\,d}x}=\dfrac{1}{2}\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {\mathrm{e}^{2x}\mathrm{\,d}x}=2\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {\mathrm{e}^{2x}\mathrm{\,d}x}=-2\mathrm{e}^{2x}+C$} \end{ex} \begin{ex} Biết $F(x)$ nguyên hàm hàm số $f(x)=2x+1$ $F(1)=3$, tính $F(0)$ \choice {$F(0)=0$} {$F(0)=5$} {\True $F(0)=1$} {$F(0)=3$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=x\ln (x+2)$ \choice {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=\dfrac{x^2}{2}\ln (x+2)-\dfrac{x^2+4x}{4}+C$} {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x}=\dfrac{x^2-4}{2}\ln (x+2)-\dfrac{x^2-4x}{4}+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=\dfrac{x^2}{2}\ln (x+2)-\dfrac{x^2+4x}{2}+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=\dfrac{x^2-4}{2}\ln (x+2)-\dfrac{x^2+4x}{2}+C$} \end{ex} \textbf{SỞ LÂM ĐỒNG} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ \choice {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x}=\dfrac{1}{2}\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x}=\mathrm{e}^{2x}\ln 2+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x}=\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x}=2\mathrm{e}^{2x}+C$} \end{ex} \textbf{SỞ NINH BÌNH} \\ \begin{ex} Mệnh đề đúng? \choice {$\displaystyle\int\limits {\mathrm{e}^{4x} \mathrm{\,d}x}=4\mathrm{e}^{4x}+C$} {$\displaystyle\int\limits {\mathrm{e}^{4x} \mathrm{\,d}x}=\dfrac{\mathrm{e}^{4x+1}}{4x+1}+C$} {\True $\displaystyle\int\limits {\mathrm{e}^{4x} \mathrm{\,d}x}=\dfrac{\mathrm{e}^{4x}}{4}+C$} {$\displaystyle\int\limits {\mathrm{e}^{4x} \mathrm{\,d}x}=\mathrm{e}^{4x}+C$} \end{ex} \begin{ex} Hàm số \textbf{không} nguyên hàm hàm số $f(x)=\dfrac{x(2+x)}{(x+1)^2}$ ? \choice {$\dfrac{x^2-x-1}{x+1}$} {$\dfrac{x^2+x+1}{x+1}$} {\True $\dfrac{x^2+x-1}{x+1}$} {$\dfrac{x^2}{x+1}$} \end{ex} \textbf{SỞ HẢI DƯƠNG} \\ \begin{ex} Tìm họ nguyên hàm hàm số $f(x)=\sin 2x$ \choice {$\displaystyle\int\limits {\sin 2x}\mathrm{\,d}x=-2\cos 2x+C$} {$\displaystyle\int\limits {\sin 2x}\mathrm{\,d}x=-\dfrac{1}{2}\cos 2x+C$} {$\displaystyle\int\limits {\sin 2x}\mathrm{\,d}x=2\cos 2x+C$} {$\displaystyle\int\limits {\sin 2x}\mathrm{\,d}x=\dfrac{1}{2}\cos 2x+C$} \end{ex} \begin{ex} Cho hai hàm số $f(x),g(x)$ hàm số liên tục $R$, có $F(x),G(x)$ nguyên hàm $f(x),g(x)$ Xét mệnh đề sau: \\ $(I):$ $F(x)+G(x)$ nguyên hàm $f(x)+g(x)$ \\ $(II):$ $k.F(x)$ nguyên hàm $kf(x)\left(k\in R\right)$ \\ $(III):$ $F(x).G(x)$ nguyên hàm $f(x).g(x)$ \\ Những mệnh đề mệnh đề đúng? \choice {$(I)$ $(II)$} {$(I),(II)$ $(III)$} {$(II)$} {$(I)$} \end{ex} \begin{ex} Cho hàm số $f(x)=2x+\sin x+2\cos x$ Tìm nguyên hàm $F(x)$ hàm số $f(x)$ thỏa mãn $F(0)=1$ \choice {$x^2+\cos x+2\sin x-2$} {$2+\cos x+2\sin x$} {$x^2-\cos x+2\sin x$} {$x^2-\cos x+2\sin x+2$} \end{ex} {$a=1;b=-3;c=2$} {$a=1;b=-1;c=1$} \end{ex} \textbf{SỞ TPHCM CỤM 5} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\dfrac{1}{\sin^22x}$ \choice {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=2\cot 2x+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=\dfrac{1}{2}\cot 2x+C}$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x=-2\cot 2x+C}$} {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x=-\dfrac{1}{2}\cot 2x+C}$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=x.\mathrm{e}^x$ \choice {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=x^2\mathrm{e}^x+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=x\mathrm{e}^x+C$} {$\displaystyle\int\limits {f(x) \mathrm{\,d}x}=(x+1)\mathrm{e}^x+C$} {\True $\displaystyle\int\limits {f(x) \mathrm{\,d}x}=(x-1)\mathrm{e}^x+C$} \end{ex} \textbf{SỞ TPHCM CỤM 6} \\ \begin{ex} Tìm nguyên hàm $F(x)$ hàm số $f(x)=\mathrm{e}^x\left(1-3\mathrm{e}^{-2x}\right)$ \choice {$F(x)=\mathrm{e}^x-3\mathrm{e}^{-3x}+C$} {\True $F(x)=\mathrm{e}^x+3\mathrm{e}^{-x}+C$} {$F(x)=\mathrm{e}^x-3\mathrm{e}^{-x}+C$} {$F(x)=\mathrm{e}^x+3\mathrm{e}^{-2x}+C$} \end{ex} \begin{ex} Gọi $F(x)$ nguyên hàm hàm số $f(x)=\cos 5x\cos x$ thỏa mãn $F\left(\dfrac{\pi}{3}\right)=0$ Tính $F\left(\dfrac{\pi}{6}\right)$ \choice {$\dfrac{\sqrt{3}}{12}$} {$0$} {\True $\dfrac{\sqrt{3}}{8}$} {$\dfrac{\sqrt{3}}{6}$} \end{ex} \begin{ex} Gọi $F(x)=(ax^3+bx^2+cx+d)\mathrm{e}^x$ nguyên hàm hàm số $f(x)=(2x^3+9x^22x+5)\mathrm{e}^x$ Tính $a^2+b^2+c^2+d^2$ \choice {$244$} {$247$} {$245$} {\True $246$} \end{ex} \textbf{SỞ TPHCM CỤM 7} \\ \begin{ex} Cho biết $F(x)$ nguyên hàm hàm số $f(x)$ Tìm $I=\displaystyle\int\limits {\left[3f(x) +1\right] \mathrm{\,d}x}$ \choice {$I=3F(x)+1+C$} {$I=3xF(x)+1+C$} {$I=3xF(x)+x+C$} {\True $I=3F(x)+x+C$} \end{ex} \begin{ex} Tìm $\displaystyle\int\limits {\dfrac{\mathrm{\,d}x}{2x+1}}$, ta được: \choice {$\dfrac{1}{2}\ln (2x+1)+C$} {$-\dfrac{2}{(2x+1)^2}+C$} {$\ln |2x+1|+C$} {\True $\dfrac{1}{2}\ln |2x+1|+C$} \end{ex} \textbf{SỞ TPHCM CỤM 8} \\ \begin{ex} Trong khẳng định sau, khẳng định sai? \choice {$\displaystyle\int\limits {\mathrm{\,d}x=x+2C}$ (C số)} {\True $\displaystyle\int\limits {x^n \mathrm{\,d}x=\dfrac{x^{n+1}}{n+1}+C}$ (C số; $n\in \mathbb{Z}$ )} {$\displaystyle\int\limits {0 \mathrm{\,d}x=C}$ (C số)} {$\displaystyle\int\limits {\mathrm{e}^x \mathrm{\,d}x=\mathrm{e}^x-C}$ (C số)} \end{ex} \begin{ex} Cho $\displaystyle\int\limits {f(x) \mathrm{\,d}x=F(x)+C}$ Khi với $a\ne 0$, ta có $\displaystyle\int\limits {f(ax+b) \mathrm{\,d}x}$ \choice {$F(ax+b)+C$} {$aF(ax+b)+C$} {$\dfrac{1}{a+b}F(ax+b)+C$} {\True $\dfrac{1}{a}F(ax+b)+C$} \end{ex} \textbf{SỞ BÌNH PHƯỚC} \\ \begin{ex} Nguyên hàm hàm số $\displaystyle\int\limits {\left(x^2+\dfrac{3}{x}-2\sqrt{x}\right)\mathrm{\,d}x}$ \choice {$\dfrac{x^3}{3}+3\ln x-\dfrac{4}{3}\sqrt{x}+C$} {$\dfrac{x^3}{3}+3\ln |x|+\dfrac{4}{3}\sqrt{x^3}+C$} {$\dfrac{x^3}{3}-3\ln |x|-\dfrac{4}{3}\sqrt{x^3}+C$} {\True $\dfrac{x^3}{3}+3\ln |x|-\dfrac{4}{3}\sqrt{x^3}+C$} \end{ex} \textbf{SỞ HÀ NÔI} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\dfrac{1}{x^2}\cos \dfrac{2}{x}$ \choice {\True $\displaystyle\int\limits {\dfrac{1}{x^2}\cos \dfrac{2}{x} \mathrm{\,d}x=}-\dfrac{1}{2}\sin \dfrac{2}{x}+C$} {$\displaystyle\int\limits {\dfrac{1}{x^2}\cos \dfrac{2}{x} \mathrm{\,d}x=}-\dfrac{1}{2}\cos \dfrac{2} {x}+C$} {$\displaystyle\int\limits {\dfrac{1}{x^2}\cos \dfrac{2}{x} \mathrm{\,d}x=}\dfrac{1}{2}\sin \dfrac{2}{x} +C$} {$\displaystyle\int\limits {\dfrac{1}{x^2}\cos \dfrac{2}{x} \mathrm{\,d}x=}\dfrac{1}{2}\cos \dfrac{2}{x} +C$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ \choice {\True $\displaystyle\int\limits {\mathrm{e}^{2x} \mathrm{\,d}x=\dfrac{1}{2}\mathrm{e}^{2x}}+C$} {$\displaystyle\int\limits {\mathrm{e}^{2x} \mathrm{\,d}x=\mathrm{e}^{2x}}+C$} {$\displaystyle\int\limits {\mathrm{e}^{2x} \mathrm{\,d}x=2\mathrm{e}^{2x}}+C$} {$\displaystyle\int\limits {\mathrm{e}^{2x} \mathrm{\,d}x=\dfrac{\mathrm{e}^{2x+1}}{2x+1}}+C$} \end{ex} \textbf{SỞ THÁI BÌNH} \\ \begin{ex} Tính $\displaystyle\int\limits {\dfrac{1}{2x+3}\mathrm{\,d}x}$ \choice {$\dfrac{1}{2}\ln (2x+3)+C$} {$2\ln |2x+3|+C$} {$\ln |2x+3|+C$} {\True $\dfrac{1}{2}\ln |2x+3|+C$} \end{ex} \begin{ex} Nguyên hàm hàm số $f(x)=\sin 2x$ \choice {\True $-\dfrac{1}{2}\cos 2x+C$} {$2\cos 2x+C$} {$-2\cos 2x+C$} {$\dfrac{1}{2}\cos 2x+C$} \end{ex} \begin{ex} Hàm số $f(x)=\sqrt{2x+1}$ nguyên hàm hàm số sau đây: \choice {$\dfrac{1}{2\sqrt{2x+1}}$} {\True $\dfrac{1}{\sqrt{2x+1}}$} {$\dfrac{3}{2}\sqrt{(2x+1)^3}$} {$\dfrac{2}{3}\sqrt{(2x+1)^3}$} \end{ex} \begin{ex} Nếu $\displaystyle\int\limits {f(x)\mathrm{\,d}x=x\mathrm{e}^x}$ $f(x)$ bằng: \choice {$x\mathrm{e}^x$} {$x(1+\mathrm{e}^x)$} {$\mathrm{e}^x$} {\True $(1+x)\mathrm{e}^x$} \end{ex} \begin{ex} Nguyên hàm $F(x)$ hàm số $f(x)=2x^2+x^3-4$ thỏa mãn điều kiện $F(0)=0$ \choice {\True $\dfrac{x^4}{4}+\dfrac{2}{3}x^3-4x$} {$2x^3-4x^4$} {$x^3-x^4+2x$} {$3x^2+4x$} \end{ex} \textbf{SỞ VĨNH PHÚC} \\ \begin{ex} Tính nguyên hàm $\displaystyle\int\limits {\cos 3x\mathrm{\,d}x}$ \choice {$-\dfrac{1}{3}\sin 3x+C$} {$-3\sin 3x+C$} {\True $\dfrac{1}{3}\sin 3x+C$} {$3\sin 3x+C$} \end{ex} \begin{ex} Biết $\displaystyle\int\limits {f(u)\mathrm{\,d}u=F(u)+C}$ Mệnh đề đúng? \choice {$\displaystyle\int\limits {f(2x-1)\mathrm{\,d}x=2F(2x-1)+C}$} {$\displaystyle\int\limits {f(2x-1)\mathrm{\,d}x=2F(x)-1+C}$} {$\displaystyle\int\limits {f(2x-1)\mathrm{\,d}x=F(2x-1)+C}$} {\True $\displaystyle\int\limits {f(2x-1)\mathrm{\,d}x=\dfrac{1}{2}F(2x-1)+C}$} \end{ex} \textbf{SỞ HÀ NAM} \\ \begin{ex} Tìm nguyên hàm hàm số $f(x)=\mathrm{e}^{2x}$ ? \choice {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x=\dfrac{1}{2}}\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=}\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=2}\mathrm{e}^{2x}+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=2}\mathrm{e}^{2x}+C$} \end{ex} \begin{ex} Biết $F(x)$ nguyên hàm hàm số $f(x)=\dfrac{1}{2x+1}$ $F(0)=\dfrac{1}{2}$ Tính $F(4)$ \choice {\True $F(4)=ln3+\dfrac{1}{2}$} {$F(4)=ln3-\dfrac{1}{2}$} {$F(4)=ln\dfrac{3}{2}-1$} {$F(4)=ln\dfrac{3}{2}+1$} \end{ex} \textbf{ĐẠI HỌC SƯ PHẠM HN 1} \\ \begin{ex} Phát biểu sau \choice {\True $\displaystyle\int\limits {\sin 2x\mathrm{\,d}x}=\dfrac{-\cos 2x}{2}+C;C\in \mathbb{R}$} {$\displaystyle\int\limits {\sin 2x\mathrm{\,d}x}=\dfrac{\cos 2x}{2}+C;C\in \mathbb{R}$} {$\displaystyle\int\limits {\sin 2x\mathrm{\,d}x}=2\cos 2x+C;C\in \mathbb{R}$} {$\displaystyle\int\limits {\sin 2x\mathrm{\,d}x}=\cos 2x+C;C\in \mathbb{R}$} \end{ex} \begin{ex} Phát biểu sau \choice {$\displaystyle\int\limits {(x^2+1)^2\mathrm{\,d}x}=\dfrac{(x^2+1)}{3}+C;C\in \mathbb{R}$} {$\displaystyle\int\limits {(x^2+1)^2\mathrm{\,d}x}=2(x^2+1)+C;C\in \mathbb{R}$} {\True $\displaystyle\int\limits {(x^2+1)^2\mathrm{\,d}x}=\dfrac{x^3}{5}+\dfrac{2x^3}{3}+x+C;C\in \mathbb{R}$} {$\displaystyle\int\limits {(x^2+1)^2\mathrm{\,d}x}=\dfrac{x^3}{5}+\dfrac{2x^3}{3}+x$} \end{ex} \textbf{ĐẠI HỌC SƯ PHẠM HN 2} \\ \begin{ex} Biết $F(x)$ nguyên hàm hàm số $f(x)=\dfrac{x}{x^2+1}$ $F(0)=1$ Tính $F(1)$ \choice {$F(1)=\ln 2+1$} {$F(1)=\dfrac{1}{2}\ln 2+1$} {$F(1)=0$} {\True $F(1)=\ln 2+2$} \end{ex} \begin{ex} Tìm nguyên hàm hàm số $f(x)=\sin 2x$ \choice {$\displaystyle\int\limits {f(x)\mathrm{\,d}x}=\dfrac{1}{2}\cos 2x+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x=-2\cos 2x+C}$} {\True $\displaystyle\int\limits {f(x)\mathrm{\,d}x}=\dfrac{-1}{2}\cos 2x+C$} {$\displaystyle\int\limits {f(x)\mathrm{\,d}x}=2\cos 2x+C$} \end{ex} \textbf{ĐẠI HỌC SƯ PHẠM HN 4} \\ \begin{ex} Hàm số sau nguyên hàm hàm số $y=\tan^2x-\cot^2x$ ? \choice {$y=\dfrac{1}{\sin x}-\dfrac{1}{\cos x}$} {$y=\tan x-\cot x$} {$y=\dfrac{1}{\sin x}+\dfrac{1}{\cos x}$} {\True $y=\tan x+\cot x$} \end{ex} \begin{ex} Tìm hàm số $F(x)$ biết $F'(x)=\dfrac{1}{\sin^2x}$ đồ thị $F(x)$ qua điểm $M\left(\dfrac{\pi}{6};\,0\right)$ \choice {$F(x)=\dfrac{1}{\sin x}+\sqrt{3}$} {$F(x)=\cot x+\sqrt{3}$} {$F(x)=\tan x+\sqrt{3}$} {\True $F(x)=-\cot x+\sqrt{3}$} \end{ex} \begin{ex} Tìm hàm $F(x)$ biết $F'(x)=3x^2-4x$ $F(0)=1$ \choice {\True $F(x)=x^3-2x^2+1$} {$F(x)=x^3-4x^2+1$} {$F(x)=\dfrac{1}{3}x^3-x^2+1$} {$F(x)=x^3+2x^2+1$} \end{ex} \textbf{ĐẠI HỌC SƯ PHẠM HN 5} \\ \begin{ex} Tìm hàm số $F(x)$ thỏa mãn điều kiện $F'(x)=\dfrac{2x^3-x}{\sqrt{x^4-x^2+1}}$ $F(0)=1$ \choice {$F(x)=\sqrt{x^4-x^2+1}+x$} {$F(x)=\sqrt{x^4-x^2+1}-x$} {\True $F(x)=\sqrt{x^4-x^2+1}$} {$F(x)=\dfrac{1}{\sqrt{x^4-x^2+1}}$} \end{ex} \textbf{ĐH VINH LẦN 1} \\ \begin{ex} Cho $F(x)$ nguyên hàm $f(x)=\mathrm{e}^{3x}$ thỏa $F(0)=1$ Mệnh đề sau đúng? \choice {$F(x)=\dfrac{1}{3}\mathrm{e}^{3x}+1$} {$F(x)=\dfrac{1}{3}\mathrm{e}^{3x}$} {\True $F(x)=\dfrac{1}{3}\mathrm{e}^{3x}+\dfrac{2}{3}$} {$F(x)=-\dfrac{1}{3}\mathrm{e}^{3x}+\dfrac{4}{3}$} \end{ex} \textbf{ĐH VINH LẦN 2} \\ \begin{ex} Mệnh đề sau đúng? \choice {$\displaystyle\int\limits {\dfrac{\mathrm{\,d}x}{\sqrt{x}}=2\sqrt{x}}+C$} {$P=4(x^2+y^2)+15xy$} {$\displaystyle\int\limits {\dfrac{\mathrm{\,d}x}{x+1}=\ln |x|+C}$} {$\displaystyle\int\limits {2^x\mathrm{\,d}x=2^x+C}$} \end{ex} \begin{ex} Biết $F(x)$ nguyên hàm hàm số $f(x)=\sin (1-2x)$ thỏa mãn $F\left(\dfrac{1} {2}\right)=1$ Mệnh đề sau đúng? \choice {$F(x)=-\dfrac{1}{2}\cos (1-2x)+\dfrac{3}{2}$} {$F(x)=\cos (1-2x)$} {$F(x)=\cos (1-2x)+1$} {\True $F(x)=\dfrac{1}{2}\cos (1-2x)+\dfrac{1}{2}$} \end{ex} \textbf{ĐH VINH LẦN 3} \\ \begin{ex} Khẳng định sau đúng? \choice {$\displaystyle\int\limits {\tan x\mathrm{\,d}x=-\ln \left|\cos x\right|+C}$} {$\displaystyle\int\limits {\sin \dfrac{x}{2}\mathrm{\,d}x=2\cos \dfrac{x}{2}+C}$} {$\displaystyle\int\limits {\cot x\mathrm{\,d}x=-\ln \left|\sin x\right|+C}$} {$\displaystyle\int\limits {\cos \dfrac{x}{2}\mathrm{\,d}x=-2\sin \dfrac{x}{2}+C}$} \end{ex} \begin{ex} Cho hàm số $y=f(x)$ thỏa mãn $f'(x)=(x+1)\mathrm{e}^x$ $\displaystyle\int\limits {f(x)}\,\mathrm{\,d}x=(ax+b)\mathrm{e}^x+c$, với $a,\,b,\,c$ số Khi đó: \choice {$a+b=2$} {$a+b=3$} {$a+b=0$} {$a+b=1$} \end{ex} \textbf{ĐH VINH LẦN 4} \\ \begin{ex} Tìm tất nguyên hàm hàm số $f(x)=-\cos 2x$ \choice {$F(x)=\dfrac{1}{2}\sin2x+C$} {\True $F(x)=-\dfrac{1}{2}\sin2x+C$} {$F(x)=-\sin2x+C$} {$F(x)=-\sin2x$} \end{ex} \begin{ex} Tìm tất nguyên hàm hàm số $f(x)=\dfrac{2}{\sqrt{x+1}}$ \choice {$F(x)=\dfrac{1}{\sqrt{x+1}}$} {$F(x)=\sqrt{x+1}$} {\True $F(x)=4\sqrt{x+1}$} {$F(x)=2\sqrt{x+1}$} \end{ex} \textbf{KHTN LẦN 1} \\ \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {\sqrt{2x+1}\mathrm{\,d}x}$ \choice {$I=\dfrac{2}{3}\sqrt{(2x+1)^3}+C$} {$I=\dfrac{1}{2\sqrt{2x+1}}+C$} {\True $I=\dfrac{1}{3}\sqrt{(2x+1)^3}+C$} {$I=\dfrac{1}{4\sqrt{2x+1}}+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {\dfrac{1+\ln x}{x}\mathrm{\,d}x}$ \choice {$I=\dfrac{1}{2}\ln^2x+\ln x+C$} {$I=\ln^2x+\ln x+C$} { I = x + ln x + C } {$I=x+\dfrac{1}{2}\ln^2x+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {\tan 2x\mathrm{\,d}x}$ \choice {$I=\dfrac{1}{2}\ln \left|\sin 2x\right|+C$} {\True $I=-\dfrac{1}{2}\ln \left|c os2x\right|+C$} {$I=2\ln \left|\sin 2x\right|+C$} {$I=-\ln \left|c os2x\right|+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {\dfrac{x\ln (x^2+1)}{x^2+1}\mathrm{\,d}x}$ \choice {$I=\ln (x^2+1)+C$} {\True $I=\dfrac{1}{4}\ln^2(x^2+1)+C$} {$I=\dfrac{1}{2}\ln (x^2+1)+C$} {$I=\ln^2(x^2+1)+C$} \end{ex} \textbf{KHTN LẦN 2} \\ \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {(2x-1)\mathrm{e}^{-x}\mathrm{\,d}x}$ \choice {$I=-(2x+1)\mathrm{e}^{-x}+C$} {$I=-(2x-1)\mathrm{e}^{-x}+C$} {$I=-(2x+3)\mathrm{e}^{-x}+C$} {$I=-(2x-3)\mathrm{e}^{-x}+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {x\ln (2x-1)\mathrm{\,d}x}$ \choice {$I=\dfrac{4x^2-1}{8}\ln |2x-1|+\dfrac{x(x+1)}{4}+C$} {$I=\dfrac{4x^2-1}{8}\ln |2x-1|-\dfrac{x(x+1)}{4}+C$} {$I=\dfrac{4x^2+1}{8}\ln |2x-1|+\dfrac{x(x+1)}{4}+C$} {$I=\dfrac{4x^2+1}{8}\ln |2x-1|-\dfrac{x(x+1)}{4}+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {(x-1)\sin 2x\mathrm{\,d}x}$ \choice {$I=\dfrac{(1-2x)\cos 2x+\sin 2x}{2}+C$} {$I=\dfrac{(2-2x)\cos 2x+\sin 2x}{2}+C$} {$I=\dfrac{(1-2x)\cos 2x+\sin 2x}{4}+C$} {$I=\dfrac{(2-2x)\cos 2x+\sin 2x}{24}+C$} \end{ex} \begin{ex} Tìm nguyên hàm $I=\displaystyle\int\limits {\dfrac{1}{4-x^2}\mathrm{\,d}x}$ \choice {$I=\dfrac{1}{2}\ln \left|\dfrac{x+2}{x-2}\right|+C$} {$I=\dfrac{1}{2}\ln \left|\dfrac{x-2}{x+2}\right|+C$} {$I=\dfrac{1}{4}\ln \left|\dfrac{x-2}{x+2}\right|+C$} {$I=\dfrac{1}{4}\ln \left|\dfrac{x+2}{x-2}\right|+C$} \end{ex} \textbf{KHTN LẦN 3} \\ \begin{ex} Biết $F(x)=(ax+b).\mathrm{e}^x$ nguyên hàm hàm số $y=(2x+3).\mathrm{e}^x$ Khi $a+b$ \choice {$2$} {$3$} {$4$} {\True $5$} \end{ex} \begin{ex} Hàm số sau làm nguyên hàm hàm số $y=2\sin 2x$ \choice {$2\sin^2x$} {$-2\cos^2x$} {$-1-\cos 2x$} {\True $-1-2\cos x\sin x$} \end{ex} \textbf{KHTN LẦN 4} \\ \begin{ex} Mệnh đề \textbf{sai}? \choice {$\displaystyle\int\limits {f'(x) \mathrm{\,d}x}=f(x)+C$ với hàm $f(x)$ có đạo hàm $\mathbb{R} $} {$\displaystyle\int\limits {kf(x) \mathrm{\,d}x}=k\displaystyle\int\limits {f(x) \mathrm{\,d}x}$ với số $k$ với hàm số $f(x)$ liên tục $\mathbb{R}$} {$\displaystyle\int\limits {\left[f(x)-g(x)\right] \mathrm{\,d}x}=\displaystyle\int\limits {f(x) \mathrm{\,d}x}-\displaystyle\int\limits {g(x) \mathrm{\,d}x}$, với hàm số $f(x),\,g(x)$ liên tục $\mathbb{R}$} {$\displaystyle\int\limits {\left[f(x)+g(x)\right] \mathrm{\,d}x}=\displaystyle\int\limits {f(x) \mathrm{\,d}x} +\displaystyle\int\limits {g(x) \mathrm{\,d}x}$, với hàm số $f(x),\,g(x)$ liên tục $\mathbb{R} $} \end{ex} \begin{ex} Tìm nguyên hàm $\displaystyle\int\limits {\dfrac{1}{1-2x}} \mathrm{\,d}x$ \choice {$\displaystyle\int\limits {\dfrac{1}{1-2x}} \mathrm{\,d}x=\dfrac{1}{2}\ln \left|\dfrac{1}{1-2x}\right| +C$} {$\displaystyle\int\limits {\dfrac{1}{1-2x}} \mathrm{\,d}x=\dfrac{1}{2}\ln |1-2x|+C$} {$\displaystyle\int\limits {\dfrac{1}{1-2x}} \mathrm{\,d}x=\ln |1-2x|+C$} {$\displaystyle\int\limits {\dfrac{1}{1-2x}} \mathrm{\,d}x=\ln \left|\dfrac{1}{1-2x}\right|+C$} \end{ex} \begin{ex} Tìm nguyên hàm $\displaystyle\int\limits {\dfrac{x+3}{x^2+3x+2} \mathrm{\,d}x}$ \choice {$\displaystyle\int\limits {\dfrac{x+3}{x^2+3x+2} \mathrm{\,d}x}=2\ln |x+2|-\ln |x+1|+C$} {$\displaystyle\int\limits {\dfrac{x+3}{x^2+3x+2} \mathrm{\,d}x}=2\ln |x+1|-\ln |x+2|+C$} {$\displaystyle\int\limits {\dfrac{x+3}{x^2+3x+2} \mathrm{\,d}x}=2\ln |x+1|+\ln |x+2|+C$} {$\displaystyle\int\limits {\dfrac{x+3}{x^2+3x+2} \mathrm{\,d}x}=\ln |x+1|+2\ln |x+2|+C$} \end{ex} \textbf{KHTN LẦN 5} \\ \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{2x^2+1}{\sqrt{x^2+1}}\mathrm{\,d}x}$ bằng: \choice {$x\sqrt{1+x^2}+C$} {$x\sqrt{1+x^2}+C$} {$\dfrac{\sqrt{1+x^2}}{x}+C$} {$\dfrac{\sqrt{1+x^2}}{x^2}+C$} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{\mathrm{\,d}x}{2\tan x+1}}$ bằng: \choice {$\dfrac{2x}{5}-\dfrac{1}{5}\ln \left|2\sin x+\cos x\right|+C$} {$\dfrac{x}{5}+\dfrac{2}{5}\ln \left|2\sin x+\cos x\right|+C$} {$\dfrac{x}{5}-\dfrac{1}{5}\ln \left|2\sin x+\cos x\right|+C$} {$\dfrac{x}{5}+\dfrac{1}{5}\ln \left|2\sin x+\cos x\right|+C$} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{x^2\sin x}{\cos^3x}\mathrm{\,d}x}$ bằng: \choice {$\dfrac{x^2}{2\cos^2x}-x\tan x-\ln \left|\cos x\right|+C$} {$\dfrac{x^2}{2\cos^2x}-x\tan x+\ln \left|\cos x\right|+C$} {$\dfrac{x^2}{2\cos^2x}+x\tan x+\ln \left|\cos x\right|+C$} {$\dfrac{x^2}{2\cos^2x}+x\tan x-\ln \left|\cos x\right|+C$} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{2x^3+1}{x(x^3-1)}\mathrm{\,d}x}$ \choice {$\ln \left|x-\dfrac{1}{x^2}\right|+C$} {$\ln \left|x^2+\dfrac{1}{x^2}\right|+C$} {$\ln \left|x+\dfrac{1}{x^2}\right|+C$} {$\ln \left|x^2-\dfrac{1}{x^2}\right|+C$} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{(x-2)^{10}}{(x+1)^{12}}\mathrm{\,d}x}$ \choice {$\dfrac{1}{11}{\left(\dfrac{x-2}{x+1}\right)}^{11}+C$} {$-\dfrac{1}{11}{\left(\dfrac{x-2}{x+1}\right)}^{11}+C$} {$\dfrac{1}{33}{\left(\dfrac{x-2}{x+1}\right)}^{11}+C$} {$\dfrac{1}{3}{\left(\dfrac{x-2}{x+1}\right)}^{11}+C$.} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{x^2-1}{x(x^2+1)\mathrm{\,d}x}}$ \choice {$\ln \left|x+\dfrac{1}{x}\right|+C$} {$\ln \left|x-\dfrac{1}{x^2}\right|+C$} {$\ln \left|x-\dfrac{1}{x}\right|+C$} {$\ln \left|x^2-\dfrac{1}{x}\right|+C$} \end{ex} \begin{ex} Nguyên hàm $\displaystyle\int\limits {\dfrac{\sin 4x}{\sin x+\cos x}\mathrm{\,d}x}$ bằng: \choice {$-\dfrac{\sqrt{2}}{3}\sin \left(3x+\dfrac{3\pi}{4}\right)+\sqrt{2}\sin \left(x+\dfrac{\pi}{4}\right)+C$} {$-\dfrac{\sqrt{2}}{3}\cos \left(3x+\dfrac{3\pi}{4}\right)-\sqrt{2}\cos \left(x+\dfrac{\pi}{4}\right)+C$} {$-\dfrac{\sqrt{2}}{3}\sin \left(3x+\dfrac{3\pi}{4}\right)-\sqrt{2}\sin \left(x+\dfrac{\pi}{4}\right)+C$} {$-\dfrac{\sqrt{2}}{3}\sin \left(3x+\dfrac{3\pi}{4}\right)+\sqrt{2}\cos \left(x+\dfrac{\pi}{4}\right)+C$} \end{ex} ... {mathrm{e}^{2x}mathrm{,d}x}=-2mathrm{e}^{2x}+C$} end {ex} egin {ex} Biết $F(x)$ nguyên hàm hàm số $f(x)=2x+1$ $F(1)=3$, tính $F(0)$ choice {$F(0)=0$} {$F(0)=5$} {True $F(0)=1$} {$F(0)=3$} end {ex} egin {ex} Tìm nguyên hàm hàm số $f(x)=xln... end {ex} egin {ex} Cho hai hàm số $f(x),g(x)$ hàm số liên tục $R$, có $F(x),G(x)$ nguyên hàm $f(x),g(x)$ Xét mệnh đề sau: \ $(I):$ $F(x)+G(x)$ nguyên hàm $f(x)+g(x)$ \ $(II):$ $k.F(x)$ nguyên hàm. .. end {ex} extbf{KHTN LẦN 3} \ egin {ex} Biết $F(x)=(ax+b).mathrm{e}^x$ nguyên hàm hàm số $y=(2x+3).mathrm{e}^x$ Khi $a+b$ choice {$2$} {$3$} {$4$} {True $5$} end {ex} egin {ex} Hàm số