Integrate sin(3x)^2cos(3x)

\int\sin\left(3x\right)^2\cos\left(3x\right)dx

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Answer

$\frac{1}{9}\sin\left(3x\right)^{3}+C_0$

Step by step solution

Problem

$\int\sin\left(3x\right)^2\cos\left(3x\right)dx$
1

Solve the integral $\int\frac{\sin\left(3x\right)^{\left(u-1\right)}\sin\left(3x\right)^2}{-3}du$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=\sin\left(3x\right) \\ du=3\cos\left(3x\right)dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{3\cos\left(3x\right)}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{u^2}{3}du$
4

Taking the constant out of the integral

$\frac{1}{3}\int u^2du$
5

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

$\frac{1}{3}\cdot\frac{u^{3}}{3}$
6

Substitute $u$ back for it's value, $\sin\left(3x\right)$

$\frac{1}{3}\cdot\frac{\sin\left(3x\right)^{3}}{3}$
7

Simplify the fraction

$\frac{1}{9}\sin\left(3x\right)^{3}$
8

Add the constant of integration

$\frac{1}{9}\sin\left(3x\right)^{3}+C_0$

Answer

$\frac{1}{9}\sin\left(3x\right)^{3}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.41 seconds

Views:

202