**Basic Integration Examples and 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^{11 }dx

**Solution :**

**∫ x ^{11 }**dx

** = (****x ^{12}/12) + c**

**Question 2 :**

Integrate the following with respect to x

∫ (1/x^{7}) dx

**Solution :**

**∫ **(1/x^{7}) dx **= ****∫ x ^{-7} dx **

** = ****x ^{(-7 + 1)}/(-7 + 1) + c**

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

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

**Question 3 :**

Integrate the following with respect to x

∫ ∛x^{4 }dx

**Solution :**

∫ ∛x^{4 }dx = ∫ x^{4/3 }dx

= x ^{[(4/3) + 1)]} / [(4/3) + 1)] + c

= x^{7/3}/(7/3) + c

= (3/7) x^{7/3 }+ c

**Question 4 :**

Integrate the following with respect to x

∫ (x^{5})^{1/8} dx

**Solution :**

∫ (x^{5})^{1/8 }dx = ∫ x^{5/8 }dx

= x^{[(5/8) + 1]}/[(5/8) + 1] + c

= x^{13/8}/(13/8) + c

= (8/13)x^{13/8 }+ c

**Question 5 :**

Integrate the following with respect to x

∫ (1/sin^{2}x) dx

**Solution :**

∫(1/sin^{2}x) dx = ∫cosec^{2}x dx

= -cot x + c

**Question 6 :**

Integrate the following with respect to x

∫ (tan x / cos x) dx

**Solution :**

∫(tan x / cos x) dx = ∫tan x (1/cos x) dx

= ∫tan x sec x dx

= sec x + c

**Question 7 :**

Integrate the following with respect to x

∫ (cos x / sin^{2} x) dx

**Solution :**

∫(cos x / sin^{2} x) dx = ∫(cosx/sinx) (1/ sinx) dx

= ∫cot x cosec x dx

= - cosec x + c

**Question 8 :**

Integrate the following with respect to x

∫ (1 / cos^{2} x) dx

**Solution :**

∫(1 / cos^{2} x) dx = ∫ sec^{2} x dx

= tan x + c

**Question 9 :**

Integrate the following with respect to x

∫ 12^{3} dx

**Solution :**

∫ 12^{3} dx = 12^{3}x + c

**Question 10 :**

Integrate the following with respect to x

∫ (x^{24}/x^{25}) dx

**Solution :**

∫ (x^{24}/x^{25}) dx = ∫ x^{24-}^{25} dx

= ∫ x^{-1} dx

= ∫ (1/x) dx

= log x + c

**Question 11 :**

Integrate the following with respect to x

∫ e^{x} dx

**Solution :**

∫ e^{x} dx = e^{x} + c

**Question 12 :**

Integrate the following with respect to x

∫ (1 + x^{2})^{-1} dx

**Solution :**

∫ (1 + x^{2})^{-1} dx = ∫ 1/(1 + x^{2}) dx

= tan^{-1}x + c

**Question 13 :**

Integrate the following with respect to x

∫ (1 - x^{2})^{-1/2} dx

**Solution :**

∫ (1 - x^{2})^{-1/2 }dx = ∫ 1/(1 - x^{2})^{1/2 }dx

= ∫ 1/√(1 - x^{2})^{ }dx

= sin^{-1} x + c

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