## Problem Statement

Calculate the indefinite integral:

$$
\int e^{-3 x}(2-9 x) d x
$$
## Solution

$$
\int e^{-3 x}(2-9 x) d x=
$$

Let's denote:

$$
\begin{aligned}
& u=2-9 x ; d u=-9 d x \\
& d v=e^{-3 x} d x ; v=-\frac{1}{3} e^{-3 x}
\end{aligned}
$$

We will use the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =(2-9 x) \cdot\left(-\frac{1}{3} e^{-3 x}\right)-\int\left(-\frac{1}{3} e^{-3 x}\right) \cdot(-9) d x= \\
& =-\frac{1}{3}(2-9 x) e^{-3 x}-3 \int e^{-3 x} d x=-\frac{1}{3}(2-9 x) e^{-3 x}-3\left(-\frac{1}{3} e^{-3 x}\right)+C= \\
& =-\frac{1}{3}(-1-9 x) e^{-3 x}+C=\frac{1}{3}(1+9 x) e^{-3 x}+C
\end{aligned}
$$

Source — “http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-13$ ” Categories: Kuznetsov Problem Book Integrals Problem 1 | Integrals

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## Problem Kuznetsov Integrals 1-14

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