oliviayychengwh

2021-12-31

Evaluate the indefinite integral.
$\int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dx$

Karen Robbins

Step 1
To evaluate the below indefinite integral.
$\int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dx$
Step 2
The given indefinite integral can be evaluate using the u-substitution method.
let $u={x}^{3}$
$⇒du=3{x}^{2}dx$
$⇒{x}^{2}dx=\frac{du}{3}$
$\therefore \int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dt=\int \mathrm{sin}\left(u\right)\left(\frac{du}{3}\right)$
$=\int \frac{\mathrm{sin}\left(u\right)}{3}du$
$=\frac{1}{3}\cdot \int \mathrm{sin}\left(u\right)du$

$=-\frac{1}{3}\mathrm{cos}\left({x}^{3}\right)+C$
Thus,
$\int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dx=-\frac{1}{3}\mathrm{cos}\left({x}^{3}\right)+C$

abonirali59

$\int {x}^{2}\mathrm{sin}\left({x}^{3}\right)dx$
We bring the expression $3\cdot {x}^{2}$ under the differential sign, i.e.:
$3\cdot {x}^{2}dx=d\left({x}^{3}\right),t={x}^{3}$
Then the original integral can be written as follows:
$\int \frac{\mathrm{sin}\left(t\right)}{3}dt$
This is a tabular integral:
$\int \frac{\mathrm{sin}\left(t\right)}{3}dt=-\frac{\mathrm{cos}\left(t\right)}{3}+C$
To write down the final answer, it remains to substitute ${x}^{3}$ instead of t.
$-\frac{\mathrm{cos}\left({x}^{3}\right)}{3}+C$

Vasquez