x+3 (x+1)^4 dx - Vidyadip

Question

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

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Answer

$\int\bigg[\frac{\text{x}+3}{(\text{x}+1)^4}\bigg]\text{dx}$
$=\int\bigg[\frac{\text{x}+1+2}{(\text{x}+1)^4}\bigg]\text{dx}$
$=\int\bigg[\frac{(\text{x}+1)}{(\text{x}+1)^4}+\frac{2}{(\text{x}+1)^4}\bigg]\text{dx}$
$=\int\frac{\text{dx}}{(\text{x}+1)^3}+2\int\frac{\text{dx}}{(\text{x}+1)^4}$
$=\int(\text{x}+1)^{-3}\text{dx}+2\int(\text{x}+1)^{-4}\text{dx}$
$=\bigg[\frac{(\text{x}+1)^{-3+1}}{-3+1}\bigg]+2\bigg[\frac{(\text{x}+1)^{-4+1}}{-4+1}\bigg]+\text{c}$
$=-\frac{1}{2}(\text{x}+1)^{-2}-\frac{2}{3}(\text{x}+1)^{-3}+\text{c}$
$=-\frac{1}{2(\text{x}+1)^2}-\frac{2}{3(\text{x}+1)^3}+\text{c}$

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