**Integration Practice Questions With Solutions :**

Here we are going to see some example problems in integration.

To find the formulas used in integration, please visit the page "Integration Formulas for Class 12"

**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

∫ cosec^{2}(5x - 7) dx

**Solution :**

∫ cosec^{2}(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

∫ sec^{2} x/5 dx

**Solution :**

∫ sec^{2} 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 - 81x^{2}) dx

**Solution :**

∫ 1/√(1 - 81x^{2}) 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 + 36x^{2}) dx

**Solution :**

∫ 1/(1 + 36x^{2}) dx

= ∫ 1/(1 + (6x)^{2}) dx

= tan^{-1}(6x)/6 + c

= (1/6) tan^{-1}(6x) + c

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