Question

x²+1/(x+1)²,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

here it is....plsss response if I'm wrong anywhere....do check out the answer

Answer 2

question is ----> $\int{\frac{(x^2+1)}{(x+1)^2}}\,dx$

first of all, we should rearrange (x² + 1) in such a way that it contains (x + 1)².
e.g., x² + 1 = x² + 1 - 2x + 2x
= (x + 1)² - 2x
= (x + 1)² - 2(x + 1) + 2

now,
$\int{\frac{x^2+1}{(x+1)^2}}\,dx\\\\\\=\int{\frac{(x+1)^2-2(x+1)+2}{(x+1)^2}}\,dx\\\\\\=\int{\frac{(x+1)^2}{(x+1)^2}-\frac{2(x+1)}{(x+1)^2}+\frac{2}{(x+1)^2}}\,dx\\\\\\=\int{1-\frac{2}{(x+1)}-2(x+1)^{-2}}\,dx\\\\\\=x-ln|x+1|+2\left[\frac{(x+1)^{-2+1}}{-2+1}\right]+C\\\\\\=x-ln|x+1|-\frac{2}{x+1}+C$