Question

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

Answer 1

$\int \frac{x+3}{(x+1{)}^4}dx$

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

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

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

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

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

$=\frac{1}{(x+1{)}^2}\left[\frac{-3x-3-4}{6(x+1)}\right]+c$

$=\frac{1}{6(x+1{)}^3}(-3x-7)+c$

$=\frac{-(3x+7)}{6(x+1{)}^3}+c$

$\therefore \int \frac{x+3}{(x+1{)}^4}dx=\frac{-(3x+7)}{6(x+1{)}^3}+c$.