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int x^2(ax + b)^-2 dx is equal to...

`int x^2(ax + b)^-2 dx` is equal to

A

`(2)/(a^(2))(x-(b)/(a)log(ax+b))+C`

B

`(2)/(a^(2))(x-(b)/(a)log(ax+b))-(x^(2))/(a(ax+b))+C`

C

`(2)/(a^(2))(x+(b)/(a)log(ax+b))+(x^(2))/(a(ax+b))+C`

D

`(2)/(a^(2))(x+(b)/(a)log(ax+b))-(x^(2))/(a(ax+b))+C`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `l=int(x^(2))/((ax+b)^(2))dx`
Put `ax+b=t rArr dx=(1)/(a)dt`
and `" "x=((t-b)/(a))`
`therefore" "l=(1)/(a^(3))int((t-b)^(2))/(t^(2))dt=(1)/(a^(3))int(1+(b^(2))/(t^(2))-(2b)/(t))dt`
`=(1)/(a^(3))(t-(b^(2))/(t)-2b log t)+C`
`=(1)/(a^(3))[ax+b-(b^(2))/(ax+b)-2blog(ax+b)]+C`
`=(1)/(a^(3))[(a^(2)x^(2)+b^(2)+2axb-b^(2))/((ax+b))-(a^(2)x^(2))/(ax+b)-2b log(ax+b)]+C`
`=(1)/(a^(3))[2ax-(a^(2)x^(2))/(ax+b)-2blog(ax+b)]+C`
`=(2)/(a^(2))[x-(b)/(a)log(ax+b)]-(x^(2))/(a(ax+b))+C`
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