What is the integral of-x^7e^{x^8-7} ?

The integral of-x^7e^{x^8-7} is -1/8 e^{x^8-7}+C

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integral of-x^7e^{x^8-7}

\int - x^{7} e^{x^{8} - 7} d x

- \frac{1}{8} e^{x^{8} - 7} + C

Solution steps

\int - x^{7} e^{x^{8} - 7} d x

Take the constant out:

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

= - \int x^{7} e^{x^{8} - 7} d x

Apply u-substitution

= - \int \frac{e ^{u^{\frac{8}{7}} - 7} u ^{\frac{1}{7}}}{7} d u

Take the constant out:

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

= - \frac{1}{7} \cdot \int e^{u^{\frac{8}{7}} - 7} u^{\frac{1}{7}} d u

Apply u-substitution

= - \frac{1}{7} \cdot \int \frac{7 e ^{v - 7}}{8} d v

Take the constant out:

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

= - \frac{1}{7} \cdot \frac{7}{8} \cdot \int e^{v - 7} d v

e^{v - 7} = e^{- 7} e^{v}

= - \frac{1}{7} \cdot \frac{7}{8} \cdot \int e^{- 7} e^{v} d v

Take the constant out:

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

= - \frac{1}{7} \cdot \frac{7}{8} e^{- 7} \cdot \int e^{v} d v

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= - \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{e ^{7}} \cdot \int e^{v} d v

Use the common integral:

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

= - \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{e ^{7}} e^{v}

Substitute back

= - \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{e ^{7}} e^{(x^{7})^{\frac{8}{7}}}

Simplify

- \frac{1}{7} \cdot \frac{7}{8} \cdot \frac{1}{e ^{7}} e^{(x^{7})^{\frac{8}{7}}} : - \frac{1}{8} e^{x^{8} - 7}

= - \frac{1}{8} e^{x^{8} - 7}

Add a constant to the solution

= - \frac{1}{8} e^{x^{8} - 7} + C

What is the integral of-x^7e^{x^8-7} ?
The integral of-x^7e^{x^8-7} is -1/8 e^{x^8-7}+C