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Integral of 5x3 e^(x4) dx

June 19, 2026Mathematics

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Answer

\frac{5}{4} e^{x^4} + C

5 steps

Alright, faith. Let's integrate this using u-substitution.

Choose a substitution for $u$ .
Let $u = x^4$ . This is a good choice because its derivative, $4x^3$ , is related to the $x^3$ term in the integrand.
Calculate $du$ .
Differentiate $u$ with respect to $x$ :
$\frac{du}{dx} = 4x^3$
Rearrange to solve for $dx$ :
$du = 4x^3 \, dx$ $x^3 \, dx = \frac{1}{4} du$
Substitute $u$ and $dx$ into the integral.
$\int 5x^3 e^{x^4} \, dx = \int 5 e^u \left(\frac{1}{4} du\right)$
Simplify and integrate with respect to $u$ .
Move the constant $\frac{5}{4}$ outside the integral.
$\frac{5}{4} \int e^u \, du$
The integral of $e^u$ is $e^u$ .
$\frac{5}{4} e^u + C$
Substitute back $u = x^4$ to express the answer in terms of $x$ .
$\frac{5}{4} e^{x^4} + C$

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