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 = )

Question

Accepted Answer

# Explanation:
## Step1: Recall 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 $v=-\int e^{-x}\mathrm{d}x=-e^{-x}$.
## Step2: Apply the integration - by - parts formula
$$
\begin{align*}
\int(7x - 3)e^{-x}\mathrm{d}x&=(7x - 3)(-e^{-x})-\int(-e^{-x})\times7\mathrm{d}x\
&=-(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x
\end{align*}
$$
## Step3: Integrate $\int e^{-x}\mathrm{d}x$
Since $\int e^{-x}\mathrm{d}x=-e^{-x}+C$, we have:
$$
\begin{align*}
-(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x&=-(7x - 3)e^{-x}+7(-e^{-x})+C\
&=-(7x - 3)e^{-x}-7e^{-x}+C\
&=(-7x + 3-7)e^{-x}+C\
&=-(7x + 4)e^{-x}+C
\end{align*}
$$

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall integration - by - parts formula

Step2: Apply the integration - by - parts formula

Step3: Integrate \(\int e^{-x}\mathrm{d}x\)

Answer: