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The value of $\int e^{x}\left(x^{5}+5 x^{4}+1\right) \cdot d x$ is
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Verified Answer
The correct answer is:
$e^{x} \cdot x^{5}+e^{x}+c$
Let $I=\int e^{x}\left(x^{5}+5 x^{4}+1\right) d x$
$$
\begin{aligned}
&=\int e^{x} x^{5} d x+5 \int e^{x} x^{4} d x+\int e^{x} d x \\
&=x^{5} e^{x}-\int 5 x^{4} e^{x} d x+5 \int e^{x} x^{4} d x+e^{x} \\
&=x^{5} e^{x}+e^{x}+c
\end{aligned}
$$
$$
\begin{aligned}
&=\int e^{x} x^{5} d x+5 \int e^{x} x^{4} d x+\int e^{x} d x \\
&=x^{5} e^{x}-\int 5 x^{4} e^{x} d x+5 \int e^{x} x^{4} d x+e^{x} \\
&=x^{5} e^{x}+e^{x}+c
\end{aligned}
$$
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