Evaluate x^2 e ^ 4 x d x - Vidyadip

Question

Evaluate $\int x^2 e ^{4 x} d x$

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Answer

$\text { Let } I =\int x^2 \cdot e ^{4 x} d x$
$=x^2 \int e ^{4 x} d x-\int\left[\frac{ d }{ d x}\left(x^2\right) \int e ^{4 x} d x\right] d x$
$=x^2 \cdot \frac{ e ^{4 x}}{4}-\int 2 x \cdot \frac{ e ^{4 x}}{4} d x$
$=\frac{x^2 \cdot e ^{4 x}}{4}-\frac{1}{2} \int x \cdot e ^{4 x} d x$
$=\frac{x^2 e ^{4 x}}{4}-\frac{1}{2}\left[x fe ^{4 x} d x-\int\left(\frac{ d }{ d x}(x) \int e ^{4 x} d x\right) d x\right]$
$=\frac{x^2 e ^{4 x}}{4}-\frac{1}{2}\left[x \cdot \frac{ e ^{4 x}}{4}-\int 1 \cdot \frac{ e ^{4 x}}{4} d x\right]$
$=\frac{x^2 e ^{4 x}}{4}-\frac{1}{2}\left[\frac{x \cdot e ^{4 x}}{4}-\frac{1}{4} \int e ^{4 x} d x\right]$
$=\frac{x^2 e ^{4 x}}{4}-\frac{1}{2}\left[\frac{x e ^{4 x}}{4}-\frac{1}{4} \cdot \frac{ e ^{4 x}}{4}\right]+ c $
$=\frac{x^2 e ^{4 x}}{4}-\frac{x e ^{4 x}}{8}+\frac{ e ^{4 x}}{32}+ c$
$\therefore I =\frac{ e ^{4 x}}{4}\left[x^2-\frac{x}{2}+\frac{1}{8}\right]+ c $

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