What is the integral of 6x^5e^{2x^6} ?

The integral of 6x^5e^{2x^6} is 1/2 e^{2x^6}+C

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integral of 6x^5e^{2x^6}

\int 6 x^{5} e^{2 x^{6}} d x

\frac{1}{2} e^{2 x^{6}} + C

Solution steps

\int 6 x^{5} e^{2 x^{6}} d x

Take the constant out:

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

= 6 \cdot \int x^{5} e^{2 x^{6}} d x

Apply u-substitution

: \int \frac{e ^{2 u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

= 6 \cdot \int \frac{e ^{2 u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

Take the constant out:

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

= 6 \cdot \frac{1}{5} \cdot \int e^{2 u^{\frac{6}{5}}} u^{\frac{1}{5}} d u

Apply u-substitution

: \int \frac{5 e ^{2 v}}{6} d v

= 6 \cdot \frac{1}{5} \cdot \int \frac{5 e ^{2 v}}{6} d v

Take the constant out:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \int e^{2 v} d v

Apply u-substitution

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{2} \cdot \int e^{w} d w

Use the common integral:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{2} e^{w}

Substitute back

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{2} e^{2 (x^{5})^{\frac{6}{5}}}

Simplify

6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{2} e^{2 (x^{5})^{\frac{6}{5}}} : \frac{1}{2} e^{2 x^{6}}

= \frac{1}{2} e^{2 x^{6}}

Add a constant to the solution

= \frac{1}{2} e^{2 x^{6}} + C

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What is the integral of 6x^5e^{2x^6} ?
The integral of 6x^5e^{2x^6} is 1/2 e^{2x^6}+C