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Integration by substitution Calculator

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1

Solved example of integration by substitution

$\int\left(x\cdot\cos\left(2x^2+3\right)\right)dx$
2

Solve the integral $\int x\cos\left(2x^2+3\right)dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=2x^2+3 \\ du=4xdx\end{matrix}$
3

Isolate $dx$ in the previous equation

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

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{\cos\left(u\right)}{4}du$
5

Take the constant out of the integral

$\frac{1}{4}\int\cos\left(u\right)du$
6

Divide $1$ by $4$

$\frac{1}{4}\int\cos\left(u\right)du$
7

Apply the integral of the cosine function

$\frac{1}{4}\sin\left(u\right)$
8

Substitute $u$ back for it's value, $2x^2+3$

$\frac{1}{4}\sin\left(2x^2+3\right)$
9

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$\frac{1}{4}\sin\left(2x^2+3\right)+C_0$

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