Evalute : x^2 e^ 4 x d x - Vidyadip

Question

Evalute : $\int x^2 e^{4 x} d x$

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Answer

$
\begin{aligned}
& \int x^2 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{1}{4} x^2 e^{4 x}-\frac{1}{2} \int x e^{4 x} d x \\
& =\frac{1}{4} x^2 e^{4 x}-\frac{1}{2}\left[x \int e^{4 x} d x-\int\left\{\frac{d}{d x}(x) \int e^{4 x} d x\right\} d x\right] \\
& =\frac{1}{4} x^2 e^{4 x}-\frac{1}{2}\left[x \cdot \frac{e^{4 x}}{4}-\int 1 \cdot \frac{e^{4 x}}{4} d x\right] \\
& =\frac{1}{4} x^2 e^{4 x}-\frac{1}{8} x \cdot e^{4 x}+\frac{1}{8} \int e^{4 x} d x \\
& =\frac{1}{4} x^2 e^{4 x}-\frac{1}{8} x e^{4 x}+\frac{1}{8} \cdot \frac{e^{4 x}}{4}+c \\
& =\frac{1}{4} e^{4 x}\left[x^2-\frac{x}{2}+\frac{1}{8}\right]+c
\end{aligned}
$

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