INTEGRATION PRACTICE QUESTIONS WITH SOLUTIONS

Question 1 :

Integrate the following with respect to x

∫ (x + 5)dx

Solution :

∫ (x + 5)dx  =  (x + 5)(6+1)/(6 +1) + c

=  (x + 5)7/7 + c

Question 2 :

Integrate the following with respect to x

∫ 1/(2 - 3x)4 dx

Solution :

∫ 1/(2 - 3x)4 dx  =  ∫ (2 - 3x)-4 dx

=  (2 - 3x)(-4 + 1) / (-4 + 1) ⋅ (-3) + c

=  (2 - 3x)-3 / (-3) (-3) + c

=  (1/9) [1/(2 - 3x)3] + c

Question 3 :

Integrate the following with respect to x

 √(3x + 2) dx

Solution :

∫ √(3x + 2) dx  =  ∫ (3x + 2)1/2 dx

=  (3x + 2)3/2 / (3/2) (3) + c

=   (3x + 2)3/2 / (9/2) + c

=  (2/9)(3x + 2)3/2 + c

Question 4 :

Integrate the following with respect to x

 sin 3x dx

Solution :

 sin 3x dx  =  (- cos 3x/3) + c

Question 5 :

Integrate the following with respect to x

 cos (5 - 11x) dx

Solution :

cos (5 - 11x) dx  =  sin (5 - 11x) / (-11) + c

=  (-1/11) sin (5 - 11x) + c

Question 6 :

Integrate the following with respect to x

 cosec2(5x - 7) dx

Solution :

 cosec2(5x - 7) dx  =  -cot (5x - 7) (1/5) + c

=  (-1/5) cot (5x - 7) + c

Question 7 :

Integrate the following with respect to x

 e3x- 6 dx

Solution :

 e3x- 6 dx  =  e3x- 6/3 + c

=  (1/3)e3x- 6 + c

Question 8 :

Integrate the following with respect to x

 e8 - 7x dx

Solution :

 e8 - 7x dx  =  e8 - 7x /(-7) + c

=  (-1/7)e8-7x + c

Question 9 :

Integrate the following with respect to x

 1/(6 - 4x) dx

Solution :

 1/(6 - 4x) dx  =  (log (6 - 4x))/-4 + c

=  (-1/4) (log (6 - 4x)) + c

Question 10 :

Integrate the following with respect to x

 sec2 x/5 dx

Solution :

 sec2 x/5 dx  =  tan (x/5)/(1/5) + c

=  5 tan (x/5) + c

Question 11 :

Integrate the following with respect to x

 cosec (5x + 3) cot (5x + 3) dx

Solution :

 cosec (5x + 3) cot (5x + 3) dx  =  [- cosec (5x + 3)]/5 + c

 =   (-1/5) cosec (5x + 3) + c

Question 12 :

Integrate the following with respect to x

 30 sec (2 - 15x) tan (2 - 15x) dx

Solution :

 30 sec (2 - 15x) tan (2 - 15x) dx 

=  30 sec (2 - 15x)/(-15) + c

=  -2 sec (2 - 15x) + c

Question 13 :

Integrate the following with respect to x

∫ 1/√(1 - (4x)2) dx

Solution :

 1/√(1 - (4x)2) dx

  =  sin-1(4x)/4 + c 

  =  (1/4)sin-1(4x) + c 

Question 14 :

Integrate the following with respect to x

∫ 1/√(1 - 81x2) dx

Solution :

 1/√(1 - 81x2) dx

  =   1/√(1 - (9x)2) dx

=  sin-1 (9x) / 9 + c

(1/9) sin-1(9x) + c

Question 15 :

Integrate the following with respect to x

∫ 1/(1 + 36x2) dx

Solution :

 1/(1 + 36x2) dx

  =   1/(1 + (6x)2) dx

=  tan-1(6x)/6 + c

=  (1/6) tan-1(6x) + c

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