Helen Lewis

2021-12-28

Find the indefinite integral.
$\int x{e}^{1-{x}^{2}}dx$

otoplilp1

Step 1
Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.
Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.
Step 2
The given integrand is $\int x{e}^{1-{x}^{2}}dx$, consider $1-{x}^{2}$ as t and differentiate the function with respect to x
$t=1-{x}^{2}$
$\frac{dt}{dx}=\frac{d\left(1-{x}^{2}\right)}{dx}$
=0-2x
=-2x
$\frac{dt}{-2}=xdx$...(1)
Step 3
Now, substitute the value of $1-{x}^{2}$ as t and value of xdx from equation (1) in $\int x{e}^{1-{x}^{2}}dx$
$\int x{e}^{1-{x}^{2}}=\int {e}^{t}\frac{dt}{-2}$
$=\frac{-1}{2}\int {e}^{t}dt$
$=\frac{-1}{2}{e}^{t}+C$
$=\frac{-1}{2}\left({e}^{1-{x}^{2}}\right)+C$ (As $t=1-{x}^{2}$)
Therefore, value of $\int x{e}^{1-{x}^{2}}dx$ is equal to $\frac{-1}{2}\left({e}^{1-{x}^{2}}\right)+C$.

trisanualb6

$\int x\cdot {e}^{1-{x}^{2}}dx$
We put the expression 2 * x under the differential sign, i.e.:
$2xdx=d\left({x}^{2}\right),t={x}^{2}$
Then the original integral can be written as follows:
$\int \frac{{e}^{1-t}}{2}dt$
This is a tabular integral:
$\int {e}^{1-t}2dt=-\frac{{e}^{1-t}}{2}+C$
To write down the final answer, it remains to substitute t instead ${x}^{2}$.
$-\frac{{e}^{1}-{x}^{2}}{2}+C$

Vasquez