Integrate e^xx*3

\int3\cdot e^x\cdot xdx

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Answer

$-3e^x+3xe^x+C_0$

Step by step solution

Problem

$\int3\cdot e^x\cdot xdx$
1

Taking the constant out of the integral

$3\int xe^xdx$
2

Use the integration by parts theorem to calculate the integral $\int xe^xdx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^xdx}\\ \displaystyle{\int dv=\int e^xdx}\end{matrix}$
5

Solve the integral

$v=\int e^xdx$
6

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$3\int xe^xdx$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$3\left(xe^x-\int e^xdx\right)$
8

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$3\left(xe^x-e^x\right)$
9

Multiply $\left(xe^x+-e^x\right)$ by $3$

$3xe^x-3e^x$
10

Add the constant of integration

$-3e^x+3xe^x+C_0$

Answer

$-3e^x+3xe^x+C_0$

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Problem Analysis

Main topic:

Integration by parts

Time to solve it:

0.23 seconds

Views:

204