DECOMPOSITION METHOD FOR INTEGRATION

Decomposition Method :

Sometimes it is very difficult to integrate the given function directly. But it can be integrated after decomposing it into a sum or difference of number of functions whose integrals are already known.

In most of the cases the given integrand will be any one of the algebraic, trigonometric or exponential forms, and sometimes combinations of these functions.

Example 1:

Integrate (1 + x²)³ dx 

Solution :

First let us expand the given expression using the formula (a + b)³

(a + b)³  =  a³ + 3a² b + 3ab² + b³

(1+ x²)³  =  1  + 3x² + 3x⁴ + x⁶ 

=  ∫dx + 3∫x²dx + 3∫x⁴ dx + ∫ x⁶ dx

=  ∫ dx + 3 ∫(x²) dx + 3 ∫ (x⁴) dx + ∫ x⁶ dx

=  x + 3 (x³/3) + 3 x⁵/5 + x⁷/7 + c

=  x + x³ + (3/5) x⁵ + (1/7) x⁷+ c

Example 2:

Integrate (tan x + cot x)² dx 

Solution :

(a + b)²  =  a² + 2ab + b²

(tan x + cot x)²  =  tan²x + cot²x + 2 tan x cot x

  =  tan²x + cot²x + 2 tan x (1/tan x)

  =  tan²x + cot²x + 2 

   =  ∫ (tan²x  + cot²x  + 2)  dx

  =  ∫ (sec²x - 1  + cosec²x - 1  + 2)  dx

  =  ∫(sec²x + cosec²x)  dx

  =  ∫sec²x dx + ∫cosec²x dx 

  =  tan x -cot x + c 

Example 3:

Integrate √(1 + sin 2x) dx 

Solution :

∫√(1 + sin 2x) dx 

1  =  sin²x + cos²x 

  =  ∫√(sin²x + cos²x + 2 sin x cos x) dx 

  =  ∫√(sin x + cos x)² dx 

  =  ∫(sin x + cos x) dx 

  =  ∫sin x dx + ∫cos x dx

  =  - cos x + sin x + c

Example 4 :

Integrate the following functions with respect to x :

(√x + (1/√x))²

Solution :

∫ (√x + (1/√x))² dx

Expanding this using the formula (a + b)²  =  a² + 2ab + b² 

  =  ∫ [(√x)² + (1/√x)² + 2√x(1/√x)] dx

  =  ∫ x dx + ∫(1/x) dx + 2 ∫ dx

  =  (x²/2) + log x + 2 x + c

Example 5 :

Integrate the following functions with respect to x :

(2x - 5)(36 + 4x)

Solution :

∫(2x - 5)(36 + 4x) dx 

  =  ∫ (72x + 8x² - 180 - 20x) dx

  =  ∫ (8x² + 52x - 180) dx

  =  ∫ 8x² dx + ∫52x dx - 180 ∫dx

  =  (8/3)x³ + 26x² - 180 x + c

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