A) \[\frac{2}{{{a}^{2}}}\left( x-\frac{b}{a}\log (ax+b) \right)+c\]
B) \[\frac{2}{{{a}^{2}}}\left( x-\frac{b}{a}\log (ax+b) \right)-\frac{{{x}^{2}}}{a(ax+b)}+c\]
C) \[\frac{2}{{{a}^{2}}}\left( x+\frac{b}{a}\log (ax+b) \right)+\frac{{{x}^{2}}}{a(ax+b)}+c\]
D) \[\frac{2}{{{a}^{2}}}\left( x+\frac{b}{a}\log (ax+b) \right)-\frac{{{x}^{2}}}{a(ax+b)}+c\]
E) \[\frac{2}{{{a}^{2}}}\left( x-\frac{b}{a}\log (ax+b) \right)+\frac{{{x}^{2}}}{a(ax+b)}+c\]
Correct Answer: B
Solution :
Let \[I=\int{\frac{{{x}^{2}}}{{{(ax+b)}^{2}}}}dx\] Put \[ax+b=t\] \[\Rightarrow \] \[dx=\frac{1}{a}dt\]and\[x=\left( \frac{t-b}{a} \right)\] \[\therefore \] \[I=\frac{1}{{{a}^{3}}}\int{\frac{{{(t-b)}^{2}}}{{{t}^{2}}}}dt\] \[=\frac{1}{{{a}^{3}}}\int{\left( 1+\frac{{{b}^{2}}}{{{t}^{2}}}-\frac{2b}{t} \right)}dt\] \[=\frac{1}{{{a}^{3}}}\int{\left( t-\frac{{{b}^{2}}}{t}-2b\,\log t \right)}+c\] \[=\frac{1}{{{a}^{3}}}\int{\left( ax+b-\frac{{{b}^{2}}}{ax+b}-2b\log (ax+b) \right)}+c\] \[=\frac{2}{{{a}^{2}}}\left( x-\frac{b}{a}\log (ax+b) \right)-\frac{{{x}^{2}}}{a(ax+b)}+c\]
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