What is the integral of 15x^{17}e^{x^9} ?

The integral of 15x^{17}e^{x^9} is 5/3 (e^{x^9}x^9-e^{x^9})+C

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integral of 15x^{17}e^{x^9}

\int 15 x^{17} e^{x^{9}} d x

\frac{5}{3} (e^{x^{9}} x^{9} - e^{x^{9}}) + C

Solution steps

\int 15 x^{17} e^{x^{9}} d x

Take the constant out:

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

= 15 \cdot \int x^{17} e^{x^{9}} d x

Apply u-substitution

: \int \frac{e ^{u^{\frac{9}{17}}} u ^{\frac{1}{17}}}{17} d u

= 15 \cdot \int \frac{e ^{u^{\frac{9}{17}}} u ^{\frac{1}{17}}}{17} d u

Take the constant out:

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

= 15 \cdot \frac{1}{17} \cdot \int e^{u^{\frac{9}{17}}} u^{\frac{1}{17}} d u

Apply u-substitution

: \int \frac{17 e ^{v} v}{9} d v

= 15 \cdot \frac{1}{17} \cdot \int \frac{17 e ^{v} v}{9} d v

Take the constant out:

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

= 15 \cdot \frac{1}{17} \cdot \frac{17}{9} \cdot \int e^{v} v d v

Apply Integration By Parts

: e^{v} v - \int e^{v} d v

= 15 \cdot \frac{1}{17} \cdot \frac{17}{9} (e^{v} v - \int e^{v} d v)

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

= 15 \cdot \frac{1}{17} \cdot \frac{17}{9} (e^{v} v - e^{v})

Substitute back

= 15 \cdot \frac{1}{17} \cdot \frac{17}{9} (e^{(x^{17})^{\frac{9}{17}}} (x^{17})^{\frac{9}{17}} - e^{(x^{17})^{\frac{9}{17}}})

Simplify

15 \cdot \frac{1}{17} \cdot \frac{17}{9} (e^{(x^{17})^{\frac{9}{17}}} (x^{17})^{\frac{9}{17}} - e^{(x^{17})^{\frac{9}{17}}}) : \frac{5}{3} (e^{x^{9}} x^{9} - e^{x^{9}})

= \frac{5}{3} (e^{x^{9}} x^{9} - e^{x^{9}})

Add a constant to the solution

= \frac{5}{3} (e^{x^{9}} x^{9} - e^{x^{9}}) + C

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What is the integral of 15x^{17}e^{x^9} ?
The integral of 15x^{17}e^{x^9} is 5/3 (e^{x^9}x^9-e^{x^9})+C