# Integrate x(7+e^x^2)*2

## \int2x\cdot\left(7+\left(e^x\right)^2\right)dx

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$-e^{2x}-14x^2+xe^{2x}+14x^2+C_0$

## Step by step solution

Problem

$\int2x\cdot\left(7+\left(e^x\right)^2\right)dx$
1

Applying the power of a power property

$\int2x\left(e^{2x}+7\right)dx$
2

Taking the constant out of the integral

$2\int x\left(e^{2x}+7\right)dx$
3

Use the integration by parts theorem to calculate the integral $\int x\left(e^{2x}+7\right)dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
4

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
5

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\left(e^{2x}+7\right)dx}\\ \displaystyle{\int dv=\int \left(e^{2x}+7\right)dx}\end{matrix}$
6

Solve the integral

$v=\int\left(\left(e^{2x}+7\right)\right)dx$
7

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

$2\left(x\int\left(e^{2x}+7\right)dx-\int\int\left(e^{2x}+7\right)dxdx\right)$
8

The integral of a sum of two or more functions is equal to the sum of their integrals

$2\left(x\left(\int e^{2x}dx+\int7dx\right)-\left(\int\int e^{2x}dxdx+\int\int7dxdx\right)\right)$
9

The integral of a constant is equal to the constant times the integral's variable

$2\left(x\left(\int e^{2x}dx+\int7dx\right)-\left(\int\int e^{2x}dxdx+x\int7dx\right)\right)$
10

The integral of a constant is equal to the constant times the integral's variable

$2\left(x\left(\int e^{2x}dx+\int7dx\right)-\left(\int\int e^{2x}dxdx+7x\cdot x\right)\right)$
11

When multiplying exponents with same base you can add the exponents

$2\left(x\left(\int e^{2x}dx+\int7dx\right)-\left(\int\int e^{2x}dxdx+7x^2\right)\right)$
12

The integral of a constant is equal to the constant times the integral's variable

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\int\int e^{2x}dxdx+7x^2\right)\right)$
13

Solve the integral $\int\int e^{2x}dxdx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=2x \\ du=2dx\end{matrix}$
14

Isolate $dx$ in the previous equation

$\frac{du}{2}=dx$
15

Substituting $u$ and $dx$ in the integral

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\int\frac{\int e^udx}{2}du+7x^2\right)\right)$
16

Taking the constant out of the integral

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\frac{1}{2}\int\int e^udxdx+7x^2\right)\right)$
17

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 $a \neq 1$

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\frac{1}{2}\int e^udx+7x^2\right)\right)$
18

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 $a \neq 1$

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\frac{1}{2}e^u+7x^2\right)\right)$
19

Substitute $u$ back for it's value, $2x$

$2\left(x\left(\int e^{2x}dx+7x\right)-\left(\frac{1}{2}e^{2x}+7x^2\right)\right)$
20

Solve the integral $\int e^{2x}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=2x \\ du=2dx\end{matrix}$
21

Isolate $dx$ in the previous equation

$\frac{du}{2}=dx$
22

Substituting $u$ and $dx$ in the integral

$2\left(x\left(\int\frac{e^u}{2}du+7x\right)-\left(\frac{1}{2}e^{2x}+7x^2\right)\right)$
23

Taking the constant out of the integral

$2\left(x\left(\frac{1}{2}\int e^udu+7x\right)-\left(\frac{1}{2}e^{2x}+7x^2\right)\right)$
24

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 $a \neq 1$

$2\left(x\left(\frac{1}{2}e^u+7x\right)-\left(\frac{1}{2}e^{2x}+7x^2\right)\right)$
25

Substitute $u$ back for it's value, $2x$

$2\left(x\left(\frac{1}{2}e^{2x}+7x\right)-\left(\frac{1}{2}e^{2x}+7x^2\right)\right)$
26

Multiply $\left(14x+e^{2x}\right)$ by $x$

$-e^{2x}-14x^2+xe^{2x}+14x^2$
27

$-e^{2x}-14x^2+xe^{2x}+14x^2+C_0$

$-e^{2x}-14x^2+xe^{2x}+14x^2+C_0$

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