BASIC INTEGRATION EXAMPLES AND SOLUTIONS

Example 1 :

Integrate the following with respect to x

∫ x¹¹ dx

Solution :

∫ x¹¹ dx =  x ^{(11 + 1)}/(11 + 1) + c

  =  (x¹²/12) + c

Example 2 :

Integrate the following with respect to x

∫ (1/x⁷) dx

Solution :

∫ (1/x⁷) dx =  ∫ x^-7 dx 

  =  x^{(-7 + 1)}/(-7 + 1) + c

=  x^-6/(-6) + c

=  (-1/6x⁶) + c

Example 3 :

Integrate the following with respect to x

∫ ∛x⁴ dx

Solution :

∫ ∛x⁴ 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

Example 4 :

Integrate the following with respect to x

∫ (x⁵)^1/8 dx

Solution :

∫ (x⁵)^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

Example 5 :

Integrate the following with respect to x

∫ (1/sin²x) dx

Solution :

∫(1/sin²x) dx  =  ∫cosec²x dx

=  -cot x + c

Example 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

Example 7 :

Integrate the following with respect to x

∫ (cos x / sin² x) dx

Solution :

∫(cos x / sin² x) dx  =  ∫(cosx/sinx) (1/ sinx) dx

=  ∫cot x cosec x dx

=  - cosec x + c

Example 8 :

Integrate the following with respect to x

∫ (1 / cos² x) dx

Solution :

∫(1 / cos² x) dx  =  ∫ sec² x dx

=  tan x + c

Example 9 :

Integrate the following with respect to x

∫ 12³ dx

Solution :

∫ 12³ dx  =  12³x + c

Example 10 :

Integrate the following with respect to x

∫ (x²⁴/x²⁵) dx

Solution :

∫ (x²⁴/x²⁵) dx  =  ∫ x^24-25 dx

=  ∫ x^-1 dx

=  ∫ (1/x) dx

=  log x + c

Example 11 :

Integrate the following with respect to x

∫ e^x dx

Solution :

∫ e^x dx  =  e^x + c

Example 12 :

Integrate the following with respect to x

∫ (1 + x²)^-1 dx

Solution :

∫ (1 + x²)^-1 dx  =  ∫ 1/(1 + x²) dx

=  tan^-1x + c

Example 13 :

Integrate the following with respect to x

∫ (1 - x²)^-1/2 dx

Solution :

∫ (1 - x²)^-1/2 dx  =  ∫ 1/(1 - x²)^1/2 dx

=  ∫ 1/√(1 - x²) dx

=  sin^-1 x + c

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