How to solve integral by partial decomposition ?

Question

∫ (2x+3)/(x+1)³ 

Answer 1

Do you remember partial fractions?  You need to convert the single fraction into a sum of fractions.  Then you can integrate each fraction.

 

2x + 3                   A                     B                        C

_______    =   ________   +    __________  +   _________

 (x + 1)³            (x + 1)             (x + 1)²                   (x + 1)³

 

 

Now equate numerators, considering the LCD on the right side.

 

2x + 3 = A(x + 1)² + B(x + 1) + C

2x + 3 = Ax² + 2Ax + A + Bx + B + C

 

 

Then,  equate coefficients. 

 

x²:  0 = A

x:   2 = 2A + B

con: 3 = A + B + C

 

This means that

 

A = 0

B = 2

C = 1

 

Plug in the values into the decomposition, then integrate.