INTEGRATION BY PARTS PRACTICE PROBLEMS

Integrate the following :

(1)  x e-x       Solution

(2)  x cos x       Solution

(3)  x cosec2x       Solution

(4)  x sec x tan x dx       Solution

(5)  2 x e3x       Solution

(6)   x2 e 2x       Solution

(7)  x5 e^x2       Solution

(8)  x tan2 x       Solution

(9)  x cos2 x       Solution

(10)  x cos 5 x cos 2 x       Solution

(11)  x2 cos 3x       Solution

(12)  cosec3x          Solution

(13)  eax cos bx       Solution

(14)  tan-1 x       Solution

(15)  (sin-1x) (e^sin-1x)/√(1 - x²)       Solution

(16)  tan-1 [(3x-x3)/(1-3x2)]       Solution

(17)  x sin-1 (x2)            Solution

Answers :

(1)  -e-x (x + 1) + C

(2)  x  sin x + cos x + C

(3)  -x cot x + log (sin x) + C

(4)  x sec x - log (sec x + tan x) + C

(5)   (2/3) e3x [x - (1/3)] + C

(6)  (1/2) (e2x)[x - (1/2)] + C

(7)  (1/2)e^x2 [x4- 2 x2+ 2]+ C

(8)  x (tan x) - log (cos x) - (x2/2) + C

(9)   (1/2) [(x²/2)+ (x/2) (sin 2 x) + (1/4) (cos 2x) + C

(10)   (1/2){(x/7)(sin 7x) + (1/49) (cos 7x)+(x/3)(sin 3x)+(1/9)(cos 3x)}+ C

(11)  (x2/3)sin 3x + (2/9)[x cos 3 x] - (2/27)[sin 3x] + C

(12)  (1/2)[-cosec x cot x] + (1/2)[log tan (x/2)] + C

(13)  [eax/(a²+b²)][a cos b x + b sin bx]  

(14)  tan-1 x- (1/2) log (1+x2) + C

(15)   e^sin-1x (sin-1x - 1) + C

(16)   3xtan-1-  (3/2) log (1+x2) + C

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