What is the integral of 9x^8e^{7x^9} ?

The integral of 9x^8e^{7x^9} is 1/7 e^{7x^9}+C

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integral of 9x^8e^{7x^9}

\int 9 x^{8} e^{7 x^{9}} d x

\frac{1}{7} e^{7 x^{9}} + C

Solution steps

\int 9 x^{8} e^{7 x^{9}} d x

Take the constant out:

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

= 9 \cdot \int x^{8} e^{7 x^{9}} d x

Apply u-substitution

: \int \frac{e ^{7 u^{\frac{9}{8}}} u ^{\frac{1}{8}}}{8} d u

= 9 \cdot \int \frac{e ^{7 u^{\frac{9}{8}}} u ^{\frac{1}{8}}}{8} d u

Take the constant out:

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

= 9 \cdot \frac{1}{8} \cdot \int e^{7 u^{\frac{9}{8}}} u^{\frac{1}{8}} d u

Apply u-substitution

: \int \frac{8 e ^{7 v}}{9} d v

= 9 \cdot \frac{1}{8} \cdot \int \frac{8 e ^{7 v}}{9} d v

Take the constant out:

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

= 9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \int e^{7 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{7} d w

= 9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \int e^{w} \frac{1}{7} d w

Take the constant out:

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

= 9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \frac{1}{7} \cdot \int e^{w} d w

Use the common integral:

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

= 9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \frac{1}{7} e^{w}

Substitute back

= 9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \frac{1}{7} e^{7 (x^{8})^{\frac{9}{8}}}

Simplify

9 \cdot \frac{1}{8} \cdot \frac{8}{9} \cdot \frac{1}{7} e^{7 (x^{8})^{\frac{9}{8}}} : \frac{1}{7} e^{7 x^{9}}

= \frac{1}{7} e^{7 x^{9}}

Add a constant to the solution

= \frac{1}{7} e^{7 x^{9}} + C

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What is the integral of 9x^8e^{7x^9} ?
The integral of 9x^8e^{7x^9} is 1/7 e^{7x^9}+C