# 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/x7dx =  ∫ 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 :

∫ ∛xdx  =  ∫ x4/3 dx

=  x [(4/3) + 1)] / [(4/3) + 1)] + c

=  x7/3/(7/3) + c

=  (3/7) x7/3 + c

Example 4 :

Integrate the following with respect to x

∫ (x5)1/8 dx

Solution :

∫ (x5)1/8 dx  =  ∫ x5/8 dx

=  x[(5/8) + 1]/[(5/8) + 1] + c

=  x13/8/(13/8) + c

=  (8/13)x13/8 + 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  =  ∫ x24-25 dx

=  ∫ x-1 dx

=  ∫ (1/x) dx

=  log x + c

Example 11 :

Integrate the following with respect to x

∫ ex dx

Solution :

∫ ex dx  =  ex + 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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