What is the integral of x^3e^{ax^4} ?

The integral of x^3e^{ax^4} is 1/(4a)e^{ax^4}+C

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integral of x^3e^{ax^4}

\int x^{3} e^{a x^{4}} d x

\frac{1}{4 a} e^{a x^{4}} + C

Solution steps

\int x^{3} e^{a x^{4}} d x

Apply u-substitution

: \int \frac{e ^{u}}{4 a} d u

= \int \frac{e ^{u}}{4 a} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4 a} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= \frac{1}{4 a} e^{u}

Substitute back

u = a x^{4}

= \frac{1}{4 a} e^{a x^{4}}

Add a constant to the solution

= \frac{1}{4 a} e^{a x^{4}} + C

\int \frac{2 ( x ^{2} + 1 )}{x ( x ^{2} + 2 )} d x

x \to 4 - lim (\frac{1}{x ^{2} - 16})

\int \frac{x}{( 1 - x ^{2} )} d x

y = ln (x^{2} - 4 x + 1), (4, 0)

\frac{1}{2 x + 8}

What is the integral of x^3e^{ax^4} ?
The integral of x^3e^{ax^4} is 1/(4a)e^{ax^4}+C