Question
Evalute : $\int(x+1)(x+2)^7(x+3) d x$

Answer

$
\begin{aligned}
& \text { Let I }=\int(x+1)(x+2)^7(x+3) d x \\
& =\int(x+2)^7(x+1)(x+3) d x \\
& =\int(x+2)^7[(x+2)-1][(x+2)+1] d x \\
& =\int(x+2)^7\left[(x+2)^2-1\right] d x \\
& =\int\left[(x+2)^9-(x+2)^7\right] d x \\
& =\int(x+2)^9 d x-\int(x+2)^7 d x \\
& =\frac{(x+2)^{10}}{10}-\frac{(x+2)^8}{8}+c
\end{aligned}
$

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