Question

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

Answer 1

$$\begin{gathered} \int \left[\frac{x+3}{{\left(x+1\right)}^4}\right]dx \\ =\int \left[\frac{x+1+2}{{\left(x+1\right)}^4}\right]dx \\ =\int \left[\frac{\left(x+1\right)}{{\left(x+1\right)}^4}+\frac{2}{{\left(x+1\right)}^4}\right]dx \\ =\int \frac{dx}{{\left(x+1\right)}^3}+2\int \frac{dx}{{\left(x+1\right)}^4} \\ =\int {\left(x+1\right)}^{-3}dx+2\int {\left(x+1\right)}^{-4}dx \\ =\left[\frac{{\left(x+1\right)}^{-3+1}}{-3+1}\right]+2\left[\frac{{\left(x+1\right)}^{-4+1}}{-4+1}\right]+\mathrm{C} \\ =-\frac{1}{2}{\left(x+1\right)}^{-2}-\frac{2}{3}{\left(x+1\right)}^{-3}+\mathrm{C} \\ =-\frac{1}{2{\left(x+1\right)}^2}-\frac{2}{3{\left(x+1\right)}^3}+\mathrm{C} \end{gathered}$$