Integral of sin(x)x*3

\frac{\int3\sin\left(x\right)\cdot xdx}{\cos\left(x\right)^4}

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Answer

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

Step by step solution

Problem

$\frac{\int3\sin\left(x\right)\cdot xdx}{\cos\left(x\right)^4}$
1

Taking the constant out of the integral

$\frac{3\int x\sin\left(x\right)dx}{\cos\left(x\right)^4}$
2

Use the integration by parts theorem to calculate the integral $\int x\sin\left(x\right)dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sin\left(x\right)dx}\\ \displaystyle{\int dv=\int \sin\left(x\right)dx}\end{matrix}$
5

Solve the integral

$v=\int\sin\left(x\right)dx$
6

Apply the integral of the sine function

$\frac{3\int x\sin\left(x\right)dx}{\cos\left(x\right)^4}$
7

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

$\frac{3\left(\int\cos\left(x\right)dx-x\cos\left(x\right)\right)}{\cos\left(x\right)^4}$
8

Apply the integral of the cosine function

$\frac{3\left(\sin\left(x\right)-x\cos\left(x\right)\right)}{\cos\left(x\right)^4}$
9

Multiplying polynomials $3$ and $-x\cos\left(x\right)+\sin\left(x\right)$

$\frac{3\sin\left(x\right)-3x\cos\left(x\right)}{\cos\left(x\right)^4}$
10

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by parts

Time to solve it:

0.24 seconds

Views:

107