## Problem Statement

Calculate the indefinite integral:

$$
\int(3 x+4) e^{3 x} d x
$$
## Solution

$$
\int(3 x+4) e^{3 x} d x=
$$

Let's denote:

$$
\begin{aligned}
& u=3 x+4 ; d u=3 d x \\
& d v=e^{3 x} d x ; v=\frac{1}{3} e^{3 x}
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$, we get:

$$
\begin{aligned}
& =(3 x+4) \cdot \frac{1}{3} e^{3 x}-\int \frac{1}{3} e^{3 x} \cdot 3 d x=\frac{1}{3} \cdot(3 x+4) e^{3 x}-\int e^{3 x} d x= \\
& =\frac{1}{3} \cdot(3 x+4) e^{3 x}-\frac{1}{3} e^{3 x}+C=\frac{1}{3} \cdot(3 x+3) e^{3 x}+C=(x+1) e^{3 x}+C
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+1-3$ »

Categories: Kuznetsov's Problem Book Integrals Problem 1 | Integrals | Problems for Checking

- Last modified: 10:03, 7 July 2009.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals $1-4$

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