evaluate using integration by parts. (use symbolic notation and fractions where needed. use ( c ) for the arbitrary constant. absorb into ( c ) as much as possible.) ( int(7 x - 3) e^{-x} d x = )

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Accepted Answer

# Explanation:
## Step1: Apply integration by parts formula
The integration by parts formula is $\int u\mathrm{d}v=uv - \int v\mathrm{d}u$. Let $u = 7x-3$ and $\mathrm{d}v=e^{-x}\mathrm{d}x$. Then $\mathrm{d}u = 7\mathrm{d}x$ and $v=-e^{-x}$.
$$
\begin{align*}
\int(7x - 3)e^{-x}\mathrm{d}x&=(7x - 3)(-e^{-x})-\int(-e^{-x})\times7\mathrm{d}x\
&=-(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x
\end{align*}
$$
## Step2: Integrate the remaining integral
We know that $\int e^{-x}\mathrm{d}x=-e^{-x}+C$. So,
$$
\begin{align*}
-(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x&=-(7x - 3)e^{-x}+7(-e^{-x})+C\
&=-7xe^{-x}+3e^{-x}-7e^{-x}+C\
&=-7xe^{-x}-4e^{-x}+C\
&=-e^{-x}(7x + 4)+C
\end{align*}
$$

# Answer:
$-e^{-x}(7x + 4)+C$

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Question

Explanation:

Step1: Apply integration by parts formula

Step2: Integrate the remaining integral

Answer: