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

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

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

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

e^{x^{6}} + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

= e^{x^{6}}

Add a constant to the solution

= e^{x^{6}} + C

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