What is the integral of (x^4e^{x^5})/5 ?

The integral of (x^4e^{x^5})/5 is 1/25 e^{x^5}+C

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integral of (x^4e^{x^5})/5

\int \frac{x ^{4} e ^{x^{5}}}{5} d x

\frac{1}{25} e^{x^{5}} + C

Solution steps

\int \frac{x ^{4} e ^{x^{5}}}{5} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int x^{4} e^{x^{5}} d x

Apply u-substitution

= \frac{1}{5} \cdot \int \frac{e ^{u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

Take the constant out:

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

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int e^{u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int \frac{4 e ^{v}}{5} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{5} \cdot \frac{1}{4} \cdot \frac{4}{5} e^{v}

Substitute back

= \frac{1}{5} \cdot \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}}

Simplify

\frac{1}{5} \cdot \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}} : \frac{1}{25} e^{x^{5}}

= \frac{1}{25} e^{x^{5}}

Add a constant to the solution

= \frac{1}{25} e^{x^{5}} + C

What is the integral of (x^4e^{x^5})/5 ?
The integral of (x^4e^{x^5})/5 is 1/25 e^{x^5}+C