INTEGRATION DECOMPOSITION METHOD

Sometimes it is difficult to integrate a 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.

In the examples given below, integrate the functions with respect to x :

Example 1 :

(x³ + 4x² - 3x + 2)/x²

Solution :

= ∫[(x³ + 4x² - 3x + 2)/x²]dx

= ∫(x³/x²)dx + 4∫(x²/x²)dx - 3∫(x/x²)dx + 2∫(1/x²)dx

= ∫xdx + 4∫dx - 3∫(1/x)dx + 2∫x^-2dx

= x²/2 + 4x - 3logx - 2x^-1

= x²/2 + 4x - 3logx - (2/x) + c

Example 2 :

(√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

= ∫xdx + ∫(1/x)dx + 2∫dx

= (x²/2) + logx + 2x + c

Example 3 :

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

Solution :

= ∫(2x - 5)(36 + 4x)dx

= ∫(72x + 8x² - 180 - 20x)dx

= ∫(8x² + 52x - 180)dx

= ∫8x²dx + ∫52xdx - 180∫dx

= (8/3)x³ + 26x² - 180x + c

Example 4 :

(cot²x + tan²x)

Solution :

= ∫(cot²x + tan²x) dx

= ∫(cosec²x - 1 + sec²x - 1)dx

= ∫cosec²xdx + ∫sec²xdx - ∫2dx

= -cotx + tanx - 2x + c

Example 5 :

(cos2x - cos2a)/(cosx - cosa)

Solution :

= ∫[(cos2x - cos2a)/(cosx - cosa)]dx

= ∫[(2cos²x - 1) - (2cos²a - 1)/(cosx - cosa)]dx

= ∫[2cos²x - 1 - 2cos²a + 1)/(cos x - cos a)]dx

= ∫[2(cos²x - cos²a)/(cosx - cosa)]dx 

= ∫[2(cosx - cosa)(cosx + cosa)/(cosx - cosa)]dx 

= 2∫(cosx + cosa)dx

= 2(sinx + xcosa) + c

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.