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

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

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integral of (x^3e^{x^4})

\int (x^{3} e^{x^{4}}) d x

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

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u^{\frac{4}{3}}} u ^{\frac{1}{3}}}{3} d u

= \int \frac{e ^{u^{\frac{4}{3}}} u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{u^{\frac{4}{3}}} u^{\frac{1}{3}} d u

Apply u-substitution

: \int \frac{3 e ^{v}}{4} d v

= \frac{1}{3} \cdot \int \frac{3 e ^{v}}{4} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{3}{4} \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{3}{4} e^{v}

Substitute back

= \frac{1}{3} \cdot \frac{3}{4} e^{(x^{3})^{\frac{4}{3}}}

Simplify

\frac{1}{3} \cdot \frac{3}{4} e^{(x^{3})^{\frac{4}{3}}} : \frac{1}{4} e^{x^{4}}

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

Add a constant to the solution

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

t^{2} \frac{d y}{d t} + y = 0

(tan^{c o s (x)} (x))^{^{'}}

y = (1 + 5 x)^{10}

\frac{d}{d x} (- x + \frac{1}{x} + 1)

\frac{d}{d x} (\frac{16}{x ( 4 - x ^{2} )})

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