1 Exercise on Integration 1.1 Substitution Use a suitable substitution to evaluate the following integral √ dx − 5x 13 tan xdx e3x + dx ex + 14 dx + ex √ x dx − x2 15 x(x2 + 2)99 dx √ x2 + x3 dx 16 √ x dx 25 − x2 xdx (1 + x2 )2 17 √ x dx 3x2 + √ dx x(1 + x) 18 √ x2 dx − x3 1 sin dx x x 19 x(x + 2)99 dx xe−x dx 20 √ (ln x)2 dx x 21 √ x x − 1dx ex dx + ex 22 √ (x + 2) x − 1dx 23 xdx √ x+9 24 x3 (1 + 3x2 ) dx 10 11 12 1.2 dx ex + e−x √ cos x √ dx x xdx 4x + Integration By Parts ln xdx ln x x x2 ln xdx xe−x dx dx x2 e−2x dx 13 sin(ln x)dx x cos xdx 14 x sin 4xdx x2 sin 2xdx 15 x cos−1 xdx (ln x)2 dx 16 tan−1 xdx sin−1 xdx 17 x99 ln xdx 10 x tan−1 xdx 18 ln x dx x101 11 ln(x + 19 x sec2 xdx 12 x sin2 xdx 20 e2x cos 3xdx 1.3 √ + x2 )dx Reduction Formula Prove the following reduction formulas In = xn eax dx; In = xn eax n − In−1 , n ≥ a a In = cosn xdx; In = sin x cosn−1 x n − + In−2 , n ≥ n n In = cos x n−2 dx; In = − In−2 , n ≥ + n n−1 sin x (n − 1) sin x n−1 In = xn cos xdx; In = xn sin x + nxn−1 cos x − n(n − 1)In−2 , n ≥ dx x 2n − ; In = − + In−1 , n ≥ n 2 n−1 −a ) 2a (n − 1)(x − a ) 2a (n − 1) √ xn dx 2xn x + a 2an √ ; In = − In−1 , n ≥ 2n + 2n + x+a In = (x2 In = (ln x)n dx; In = x(ln x)n − nIn−1 , n ≥ In = In = √ xn − xdx; In = 2n In−1 , n ≥ 2n − 3 1.4 Trigonometric Integrals Evaluate dx − cos x 10 dx cos x sin2 x sin5 x cos xdx 11 sin x cos3 x , dx + cos2 x sin 3x sin 5xdx 12 tan5 xdx cos x x cos dx 13 dx , dx sin x cos4 x cos3 xdx 14 sin 5x cos xdx sin4 xdx 15 cos x cos 2x cos 3xdx sec2 x tan2 xdx 16 cos5 x sin3 xdx sec x tan3 xdx 17 cos5 x sin4 xdx cot2 xdx 18 sin2 x cos4 xdx 1.5 Trigonometric Substitution Evaluate the following integrals by trigonometric substitution x2 dx + x2 dx (1 − x2 ) 1+x dx 1−x dx √ √ x2 16 − x2 dx dx √ x2 x2 + (1 + x2 ) x2 dx √ − x2 dx + x2 10 dx (4x2 + 1)3/2 (2x − x2 )3/2 1.6 Rational Functions Evaluate the following integrals of rational functions x2 dx − x2 x2 + 5x + , dx x4 + 5x2 + x3 dx 3+x dx (x + 1)(x2 + 1) (1 + x)2 dx + x2 10 dx + 2x − 11 x2 (x2 dx − 2)(x2 + 3) 12 x2 + , dx (x + 1)2 (x − 1) 13 x2 , dx (x2 − 3x + 2)2 14 1.7 2x3 − 4x2 − x − dx x2 − 2x − (x2 − 2x dx + 1)(x − 1)2 dx + 1)2 x(x2 x2 dx (x − 1)(x − 2)(x − 3) x2 (x2 xdx − 2x + 2) t-method Use t-substitution to evaluate the following integrals dx sin3 x dx + sin x dx + sin x − cos x dx + cos x dx sin x cos4 x cos x + dx sin x + cos x 1.8 Miscellaneous Evaluate the following integrals (ln x)2 dx x x+4 dx (x + 1)2 x(ln x)2 dx cos3 x dx sin2 x √ xdx − x2 xdx (1 + x2 )2 e2x dx + ex 26 x sin2 xdx dx x(1 + ln x) 27 √ cos2 x sin3 xdx 28 10 sin 2x dx + cos2 x 29 11 ex dx x2 30 sec3 x tan xdx 12 sin x dx cos2 x 31 √ x3 x2 + 1dx 13 x tan2 xdx 32 cos 2x sin 3xdx 14 cot x dx + sin x 33 x4 + x − dx x3 + x 15 x3 dx x2 − 34 x3 dx √ x2 + 16 dx e2x + ex − 35 dx ex − x2 4dx √ − x2 x+1 dx − 1) x2 (x 17 18 ln x dx x + ln x √ − x2 dx x2 √ (x2 dx − 1)2 36 dx √ 1+ x 37 √ cos xdx 38 tan4 xdx 19 x2 dx x2 + 20 √ dx x2 + 39 √ 21 cos3 x dx sin x 40 x2 tan−1 xdx 22 x2 + dx x2 − 5x + 41 sin−1 xdx 23 xdx √ x−2 42 24 √ dx + ex 43 25 cos(ln x)dx 44 dx x(x − 1) √ xdx √ 1−x √ x+1 dx x √ √ x − xdx Section 1.1: Substitution √ − 52 − 5x + C 2x e − ex + x + C √ − − x2 + C (1 4 + x3 ) + C − 2(1+x 2) + C tan−1 √ x+C cos x1 + C − 12 e−x + C (ln x)3 +C 13 − ln | cos x| + C 14 x − ln(1 + ex ) + C 15 (x2 200 + 2)100 + C √ 16 − 25 − x2 + C √ 17 13 3x2 + + C √ 18 − 23 − x3 + C 19 (x+2)101 101 20 (2x 12 21 (x 15 10 ln(2 + ex ) + C 22 (x 11 tan−1 ex + C √ 12 sin x + C 24 135 − (x+2)100 50 + C √ − 5) 4x + + C − 1)3/2 (3x + 2) + C − 1)3/2 (x + 4) + C √ 23 23 (x − 18) x + + C (3x2 + 1) 3/2 (9x2 − 2) + C Section 1.2: Integration By Parts x ln x − x + C 11 x ln(x + √ + x2 ) − √ + x2 + C − 13 ) + C 12 x2 − x1 ((ln x)2 + ln x + 2) + C 13 x (sin(ln x) −(x + 1)e−x + C 14 16 − e (2x2 + 2x + 1) + C 15 x2 cos−1 x x sin x + cos x + C 16 x tan−1 x − 12 log(x2 + 1) + C x3 (ln x −2x − 2x 4−1 cos 2x + x2 sin 2x + C x(ln x)2 − 2x ln x + 2x + C √ x sin−1 x + − x2 + C 10 − x2 + 1+x2 tan−1 x + C 17 − x4 sin 2x − 18 cos 2x + C − cos(ln x)) + C sin 4x − 14 x cos 4x + C x100 100 + sin−1 x √ x 1−x2 x100 10000 +C ln x 100x100 +C ln x − 18 − 10000x 100 − − +C 19 x tan x + ln(cos x) + C 20 2x e (3 sin 3x 13 + cos 3x) + C 7 Section 1.4: Trigonometric Integrals − cot x2 + C 11 − 12 cos2 x + 21 ln(1 + cos2 x) + C sin6 x + C 12 16 sin 2x − sin 8x + C x sin + sin 5x tan2 x − − ln | cos x| + C 13 −8 cot 2x − 83 cot3 2x + C +C 12 14 − 81 cos 4x − 3 tan4 sin x − sin x + C cos 6x + C 15 x tan3 x + C 16 cos8 (x) − cos6 (x) +C sec3 x + − sec x + C 17 sin9 (x) − sin7 (x) + x 8 − 14 sin 2x + 32 sin 4x + C −x − cot x + C + sin 2x + sin 4x 16 sin 6x 24 +C sin5 (x) +C + 18 − 16 cos5 x sin x + 24 cos3 x sin x + 1 cos x sin x + 16 x + C 16 x 10 − sin1 x + 21 ln 1+sin +C 1−sin x Section 1.5: Trigonometric Substitution x − tan−1 x + C √ x 1−x2 ln |x + +C √ − − x2 + sin−1 x + C √ x 1+x2 x − x √ − x2 + C + x2 | + C 16 − x2 √ − +C sin−1 √ √ x2 +4 4x √ x 4x2 +1 10 √ x−1 2x−x2 x3 − 2x +32 sin−1 x +C +C Section 1.6: Rational Functions |+C −x + 12 ln | 1+x 1−x tan−1 x + 65 ln xx2 +1 +C +4 9x − 32 x2 + 13 x3 − 27 ln |3 + x| + C x + ln(1 + x2 ) + C 4 1√ 10 x+1 √ tan−1 x + 14 ln (x+1) +C x2 +1 10 x2 + ln |x + 1| + ln |x − 3| + C 11 tan−1 x − ln | x−1 |+C x+3 √2 | − ln | x− x+ 2 √ tan−1 + 12 ln |x2 − 1| + C − x25x−6 + ln | x−1 |+C −3x+2 x−2 √x +C x−1 x +1 + ln (x−1) + C 12 2(x2 +1) 13 ln(x−3)−4 ln(x−2)+ 12 ln(x−1)+C 14 ln + ln |x| − 21 ln(x2 + 1) + C x2 x2 −2x+2 − 12 tan−1 (1 − x) + C Section 1.7: t-method x − 2cos + 12 ln | tan x2 | + C sin2 x tan x − sec x + C cos x cos3 x + (x + ln(sin x + cos x + 3)) − + ln | tan x | √2 tan−1 tan( x2 )+1 √ +C tan( x2 ) √ −x+C √ 2 tan−1 +C √1 tan−1 tan( x2 )+1 √ +C Section 1.8: Miscellaneous 1 (ln x)3 2 x (ln x)2 +C − 12 x2 ln x + 14 x2 + C √ − − x2 + C ln |x + 1| − x+1 +C 23 (x 25 x (cos(ln x) 26 x − 2) + 4(x − 2) + C √ 24 x − ln(1 + + ex ) + C + sin(ln x)) + C − 14 x sin 2x − 81 cos 2x + C x − sin1 x − sin x + C 27 −2 sin−1 e− + C − 2(1+x 2) + C 28 − ex − ln(1 + ex ) + C 29 x − ln |x| + ln |x − 1| + C sec3 x + C √ 4−x2 x ln |1 + ln x| + C 30 cos5 x − 31 cos3 x + C 31 2 x (x 10 − ln(1 + cos2 x) + C 11 −e x + C +C + 1) − (x2 15 + 1) + C 32 − 10 cos 5x − 12 cos x + C − ln |x| + 12 ln(x2 + 1) + C √ 34 13 (x2 + 4) − x2 + C 33 12 sec x + C x 2 13 − x2 + x tan x + ln cos x + C 14 − ln |1 + csc x| + C 15 x + 12 ln |x2 − 1| + C x 16 − + 13 ln |ex − 1| + C √ √ 17 − + ln x + 23 (ln x) + ln x + C √ 18 − 9−x2 x − sin−1 x +C ln |x + 1| − 14 ln |x − 1| − 2(x2x−1) + C √ √ 36 x − ln(1 + x) + C √ √ √ 37 x sin x + cos x + C 35 38 40 tan−1 x − 16 x2 + 16 ln(x2 + 1) + C √ x sin−1 x + − x2 + C √ √ √ sin−1 x − x − x + C √ √ √ x + + ln | x − 1| − ln | x + 1| + C √ √ √ sin−1 x − 14 x − x(1 − 2x) + C tan3 x − tan x + x + C √ √ 39 ln | x − 1| − ln | x + 1| + C 19 x − tan−1 x + C √ 20 ln |x + x2 + 9| + C 42 21 ln | sin x| − 12 sin2 x + C 43 22 x + 17 ln |x − 3| − 12 ln |x − 2| + C 44 41 x