What is the integral of 5x^4e^{2x^5} ?

The integral of 5x^4e^{2x^5} is 1/2 e^{2x^5}+C

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integral of 5x^4e^{2x^5}

\int 5 x^{4} e^{2 x^{5}} d x

\frac{1}{2} e^{2 x^{5}} + C

Solution steps

\int 5 x^{4} e^{2 x^{5}} d x

Take the constant out:

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

= 5 \cdot \int x^{4} e^{2 x^{5}} d x

Apply u-substitution

: \int \frac{e ^{2 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

= 5 \cdot \int \frac{e ^{2 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

= 5 \cdot \frac{1}{4} \cdot \int e^{2 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

: \int \frac{4 e ^{2 v}}{5} d v

= 5 \cdot \frac{1}{4} \cdot \int \frac{4 e ^{2 v}}{5} d v

Take the constant out:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{2 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{2} d w

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{w} \frac{1}{2} d w

Take the constant out:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} \cdot \int e^{w} d w

Use the common integral:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{w}

Substitute back

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{2 (x^{4})^{\frac{5}{4}}}

Simplify

5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{2 (x^{4})^{\frac{5}{4}}} : \frac{1}{2} e^{2 x^{5}}

= \frac{1}{2} e^{2 x^{5}}

Add a constant to the solution

= \frac{1}{2} e^{2 x^{5}} + C

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What is the integral of 5x^4e^{2x^5} ?
The integral of 5x^4e^{2x^5} is 1/2 e^{2x^5}+C