3x-114. (x22

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Solution

The integral is given as,

I= ∫ ( 3x−1 )dx ( x+2 ) 2

Use partial fraction rule to simplify the fraction.

3x−1 ( x+2 ) 2 = A ( x+2 ) + B ( x+2 ) 2 3x−1=A( x+2 )+B

Substitute x=−2then,

B=−7

Substitute x=0and B=−7 then,

A=3

Substitute the above values and integral can be written as,

I= ∫ ( 3x−1 )dx ( x+2 ) 2 =3 ∫ dx x+2 −7 ∫ dx ( x+2 ) 2

On integrating, we get

I=3log| x+2 |−7[ ( x+2 ) −1 −1 ]+C =3log| x+2 |+ 7 ( x+2 ) +C

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