Integrate x(2x^2+1)^6

\int x\left(2x^2+1\right)^6dx

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Answer

$\frac{1}{28}\left(1+2x^2\right)^{7}+C_0$

Step by step solution

Problem

$\int x\left(2x^2+1\right)^6dx$
1

Solve the integral $\int x\left(1+2x^2\right)^6dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=1+2x^2 \\ du=4xdx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{4x}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{u^6}{4}du$
4

Taking the constant out of the integral

$\frac{1}{4}\int u^6du$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{4}\cdot\frac{u^{7}}{7}$
6

Substitute $u$ back for it's value, $1+2x^2$

$\frac{1}{4}\cdot\frac{\left(1+2x^2\right)^{7}}{7}$
7

Simplify the fraction

$\frac{1}{28}\left(1+2x^2\right)^{7}$
8

Add the constant of integration

$\frac{1}{28}\left(1+2x^2\right)^{7}+C_0$

Answer

$\frac{1}{28}\left(1+2x^2\right)^{7}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.33 seconds

Views:

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