Integrate sin(2x)

\int\sin\left(2x\right)dx

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Answer

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

Step by step solution

Problem

$\int\sin\left(2x\right)dx$
1

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

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

Isolate $dx$ in the previous equation

$\frac{du}{2}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{\sin\left(u\right)}{2}du$
4

Taking the constant out of the integral

$\frac{1}{2}\int\sin\left(u\right)du$
5

Apply the integral of the sine function

$\frac{1}{2}\left(-1\right)\cos\left(u\right)$
6

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

$-\frac{1}{2}\cos\left(2x\right)$
7

Applying an identity of double-angle cosine

$-\frac{1}{2}\left(1-2\sin\left(x\right)^2\right)$
8

Multiply $\left(1+-2\sin\left(x\right)^2\right)$ by $-\frac{1}{2}$

$\sin\left(x\right)^2-\frac{1}{2}$
9

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.36 seconds

Views:

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