Evaluate:

Question:
Evaluate: $\int \frac{\mathrm{x}+3}{(\mathrm{x}+1)^{4}} \mathrm{dx}$

Solution:

Let I $=\int \frac{x+3}{(x+1)^{4}} d x$

$I=\int \frac{x+3}{(x+1)^{4}} d x$

$=\int \frac{x+1}{x+1^{4}} d x+\int \frac{2}{(x+1)^{4}} d x$

$=\int \frac{1}{(x+1)^{3}} d x+\int \frac{2}{(x+1)^{4}} d x$

$=\int(x+1)^{-3} d x+\int 2(x+1)^{-4} d x$

$=\frac{[x+1]^{-2+1}}{-3+1}+\frac{2(x+1)^{-4+1}}{-4+1}$

$=\frac{[x+1]^{-2}}{-2}+\frac{2(x+1)^{-3}}{-3}$

Hence, $I=-\frac{1}{2(x+1)^{2}}-\frac{2}{3(x+1)^{3}}+C$