Question:
Evaluate $\int \frac{x^{2}}{(x-1)^{2}} d x$
Solution:
$y=\int \frac{(x-1+1)^{2}}{(x-1)^{2}} d x$
$y=\int \frac{(x-1)^{2}+2(x-1)+1}{(x-1)^{3}} d x$
$y=\int \frac{1}{(x-1)}+2 \frac{1}{(x-1)^{2}}+\frac{1}{(x-1)^{3}} d x$
Using formula $\int \frac{1}{x} d x=\ln x$ and $\int x^{n} d x=\frac{x^{n+1}}{n+1}$
$y=\ln (x-1)+2 \frac{(x-1)^{-1}}{-1}+\frac{(x-1)^{-2}}{-2}+c$
$y=\ln (x-1)-2(x-1)^{-1}-\frac{(x-1)^{-2}}{2}+c$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.