kuvitia9f

2021-12-31

Evaluate the integrals.
$\int {x}^{3}{e}^{{x}^{4}}dx$

ambarakaq8

Expert

Step 1
Solution -
Given integral -
$y=\int {x}^{3}{e}^{{x}^{4}}dx$
Let,
$t={x}^{4}$
differentiating on both sides w.r.t x,
$\frac{dt}{dx}=4{x}^{3}$
$dt=4{x}^{3}dx$
$\frac{dt}{4}={x}^{3}dx$
Step 2
Now substituting these values in the given integral,
$y=\frac{1}{4}\int {e}^{t}dt$
$y=\frac{1}{4}\left[{e}^{t}\right]+C$
where C is the constant.

Philip Williams

Expert

Step 1
This problem can be solved using a u-substitution. Let $u={x}^{4}$. Then $du=4{x}^{3}dx$.
$\int {x}^{3}{e}^{{x}^{4}}dx=\int \frac{{e}^{u}}{4}du=\frac{{e}^{u}}{4}+c=\frac{{e}^{{x}^{4}}}{4}+c$
Result:
$\frac{{e}^{{x}^{4}}}{4}+c$

Vasquez

Expert

Step 1
Substitute P
$\int {x}^{3}{e}^{{x}^{4}}dx=\frac{1}{4}\int {e}^{u}du=\frac{1}{4}{e}^{u}+C$
Substitute back $u={x}^{4}$
$=\frac{1}{4}{e}^{{x}^{4}}+C$
Result:
$\frac{1}{4}{e}^{{x}^{4}}+C$

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