What is the integral of x^4e^{8x^5} ?

The integral of x^4e^{8x^5} is 1/40 e^{8x^5}+C

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integral of x^4e^{8x^5}

\int x^{4} e^{8 x^{5}} d x

\frac{1}{40} e^{8 x^{5}} + C

Solution steps

\int x^{4} e^{8 x^{5}} d x

Apply u-substitution

: \int \frac{e ^{8 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

= \int \frac{e ^{8 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}{4} \cdot \int e^{8 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

: \int \frac{4 e ^{8 v}}{5} d v

= \frac{1}{4} \cdot \int \frac{4 e ^{8 v}}{5} d v

Take the constant out:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{8 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{8} d w

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{w} \frac{1}{8} d w

Take the constant out:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{8} \cdot \int e^{w} d w

Use the common integral:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{8} e^{w}

Substitute back

= \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{8} e^{8 (x^{4})^{\frac{5}{4}}}

Simplify

\frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{8} e^{8 (x^{4})^{\frac{5}{4}}} : \frac{1}{40} e^{8 x^{5}}

= \frac{1}{40} e^{8 x^{5}}

Add a constant to the solution

= \frac{1}{40} e^{8 x^{5}} + C

f (x) = e^{3}

f (x) = x^{5}, (1, 1)

\frac{d}{d x} (sec (x) tan (y))

\int 81 - x^{2} d x

\frac{d}{d x} (3 + x^{10} e^{x})

What is the integral of x^4e^{8x^5} ?
The integral of x^4e^{8x^5} is 1/40 e^{8x^5}+C