MAE101 CAL v2 chapter 6 techniquse of integration

ĐẠI HỌC FPT CẦN THƠ Chapter TECHNIQUES OF INTEGRATION (Page 261-330, Calculus Volume 2) Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTENT 3.1 Integration by Parts 3.6 Numerical Intergration 3.7 Improper Integrals MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ TECHNIQUES OF INTEGRATION 3.1 Integration by Parts In this section, we will learn: How to integrate complex functions by parts MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS ò f ( x) g '( x) dx = f ( x) g ( x) - ò g ( x) f '( x) dx ▪ Let u = f(x) and v = g(x) ▪ Then, the differentials are: du = f’(x) dx and dv = g’(x) dx Thus, by the Substitution Rule, the formula for integration by parts becomes:  u dv = uv −  v du MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Evaluating both sides of Formula between a and b, assuming f’ and g’ are continuous, and using the FTC, we obtain:  b a b f ( x) g '( x) dx = f ( x) g ( x) a −  g ( x) f '( x) dx MSc Truong Van Tri b a Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Example Find I =  x sin xdx MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Example Let u = x dv = sin x dx Then, du = dx v = − cos x Using Formula 2, we have: u dv u v v du  x sin x dx =  x sin x dx = x (− cos x) −  (− cos x) dx = − x cos x +  cos x dx = − x cos x + sin x + C MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Example Evaluate I= ∫ ex sinx dx  ex does not become simpler when differentiated  Neither does sin x become simpler MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Example Nevertheless, we try choosing u = ex and dv = sin x  Then, du = ex dx and v = – cos x MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTEGRATION BY PARTS Example So, integration by parts gives: e x sin x dx = −e cos x +  e cos x dx MSc Truong Van Tri x x Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition If ò b t f ( x) dx  b − exists for every number t ≤ a, then b f ( x) dx = lim  f ( x) dx t →− t provided this limit exists (as a finite number) MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONVERGENT AND DIVERGENT Definition The improper integrals are called: ò ¥ a f ( x) dx and ò b -¥ f ( x) dx  Convergent if the corresponding limit exists  Divergent if the limit does not exist MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition ò If both ¥ a f ( x) dx and ò a -¥ f ( x) dx are convergent, then we define: ò ¥ -¥ a ¥ -¥ a f ( x) dx = ò f ( x) dx + ò f ( x) dx  Here, any real number a can be used MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example For what values of p is the integral ò ¥ 1 dx p x convergent?  Convergent if p >  Divergent if p ≤ MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Investigate the convergence of the improper integrals: x e (a) x3 dx −  (b) x e x3 dx − MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPROPER INTEGRAL OF TYPE Definition If f is continuous on [a, b) and is discontinuous at b, then ò b a t f ( x) dx = lim- ò f ( x) dx t ®b a if this limit exists (as a finite number) MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPROPER INTEGRAL OF TYPE Definition If f is continuous on (a, b] and is discontinuous at a, then ò b a b f ( x) dx = lim+ ò f ( x) dx t ®a t if this limit exists (as a finite number) MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPROPER INTEGRAL OF TYPE Definition The improper integral ò b a f ( x) dx is called:  Convergent if the corresponding limit exists  Divergent if the limit does not exist MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPROPER INTEGRAL OF TYPE Definition If f has a discontinuity at c, where a < c < b, and ò both c a define: ò f ( x) dx and ò b a b c f ( x) dx are convergent, then we c b a c f ( x) dx = ò f ( x) dx + ò f ( x) dx MSc Truong Van Tri Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPROPER INTEGRAL OF TYPE Example Let b> Investigate the convergence of the improper integrals: b dx a ( x − a) p Answer: diverges if p  and converges if p