Alfred Martin

2022-01-02

Use the methods introduced evaluate the following integrals.
$\int {x}^{2}\mathrm{cos}xdx$

MoxboasteBots5h

Step 1
Given the integral $\int {x}^{2}\mathrm{cos}xdx$.
We need to evaluate the above integral.
Step 2
Let $I=\int \left({x}^{2}\mathrm{cos}x\right)dx$
We need to integrate the above integral using integrating by parts.
$u={x}^{2}$ and $v=\mathrm{cos}x$
So $\int \left(uv\right)dx=u\int vdx-\int \left[{u}^{\prime }\int vdx\right]$
$I={x}^{2}\mathrm{sin}x-\int 2x\mathrm{sin}xdx$
$I={x}^{2}\mathrm{sin}x-2\int \left(x\mathrm{sin}x\right)dx$
where

$=-x\mathrm{cos}x+\int \mathrm{cos}xdx$
$=-x\mathrm{cos}x+\mathrm{sin}x$
$I={x}^{2}\mathrm{sin}x-2\left[-x\mathrm{cos}x+\mathrm{sin}x\right]+C$
$I={x}^{2}\mathrm{sin}x+2x\mathrm{cos}x-2\mathrm{sin}x+C$

Mary Herrera

$\int {x}^{2}\cdot \mathrm{cos}\left(x\right)dx$
Prepare for integration by parts
$u={x}^{2}$
$dv=\mathrm{cos}\left(x\right)dx$
Calculate the differential
du=2xdx
$v=\mathrm{sin}\left(x\right)$
Substitute the values into the formula
${x}^{2}\cdot \mathrm{sin}\left(x\right)-\int \mathrm{sin}\left(x\right)\cdot 2xdx$
${x}^{2}\cdot \mathrm{sin}\left(x\right)-2\cdot \int \mathrm{sin}\left(x\right)\cdot xdx$
${x}^{2}\cdot \mathrm{sin}\left(x\right)-2\cdot \int x\cdot \mathrm{sin}\left(x\right)dx$
Use the partial integration
${x}^{2}\cdot \mathrm{sin}\left(x\right)-2\left(x\cdot \left(-\mathrm{cos}\left(x\right)\right)-\int -\mathrm{cos}\left(x\right)dx\right)$
${x}^{2}\cdot \mathrm{sin}\left(x\right)-2\left(x\cdot \left(-\mathrm{cos}\left(x\right)\right)+\int \mathrm{cos}\left(x\right)dx\right)$
${x}^{2}\cdot \mathrm{sin}\left(x\right)-2\left(x\cdot \left(-\mathrm{cos}\left(x\right)\right)+\mathrm{sin}\left(x\right)\right)$
Simplify
${x}^{2}\cdot \mathrm{sin}\left(x\right)+2x\cdot \mathrm{cos}\left(x\right)-2\mathrm{sin}\left(x\right)$
${x}^{2}\cdot \mathrm{sin}\left(x\right)+2x\cdot \mathrm{cos}\left(x\right)-2\mathrm{sin}\left(x\right)+C$

Vasquez