INTEGRATION PRACTICE QUESTIONS WITH SOLUTIONS

Question 1 :

Integrate the following with respect to x

∫ (x + 5)⁶ dx

Solution :

∫ (x + 5)⁶ 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

∫ e^{3x- 6} dx

Solution :

∫ e^{3x- 6} dx  =  e^{3x- 6}/3 + c

=  (1/3)e^{3x- 6} + c

Question 8 :

Integrate the following with respect to x

∫ e^{8 - 7x} dx

Solution :

∫ e^{8 - 7x} dx  =  e^{8 - 7x} /(-7) + c

=  (-1/7)e^8-7x + 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

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.