What is the integral of 50x^9e^{5x^5} ?

The integral of 50x^9e^{5x^5} is 2/5 (5e^{5x^5}x^5-e^{5x^5})+C

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integral of 50x^9e^{5x^5}

\int 50 x^{9} e^{5 x^{5}} d x

\frac{2}{5} (5 e^{5 x^{5}} x^{5} - e^{5 x^{5}}) + C

Solution steps

\int 50 x^{9} e^{5 x^{5}} d x

Take the constant out:

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

= 50 \cdot \int x^{9} e^{5 x^{5}} d x

Apply u-substitution

: \int \frac{e ^{5 u^{\frac{5}{9}}} u ^{\frac{1}{9}}}{9} d u

= 50 \cdot \int \frac{e ^{5 u^{\frac{5}{9}}} u ^{\frac{1}{9}}}{9} d u

Take the constant out:

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

= 50 \cdot \frac{1}{9} \cdot \int e^{5 u^{\frac{5}{9}}} u^{\frac{1}{9}} d u

Apply u-substitution

: \int \frac{9 e ^{5 v} v}{5} d v

= 50 \cdot \frac{1}{9} \cdot \int \frac{9 e ^{5 v} v}{5} d v

Take the constant out:

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

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \int e^{5 v} v d v

Apply u-substitution

: \int \frac{e ^{w} w}{25} d w

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \int \frac{e ^{w} w}{25} d w

Take the constant out:

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

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \frac{1}{25} \cdot \int e^{w} w d w

Apply Integration By Parts

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

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \frac{1}{25} (e^{w} w - \int e^{w} d w)

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

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \frac{1}{25} (e^{w} w - e^{w})

Substitute back

= 50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \frac{1}{25} (e^{5 (x^{9})^{\frac{5}{9}}} \cdot 5 (x^{9})^{\frac{5}{9}} - e^{5 (x^{9})^{\frac{5}{9}}})

Simplify

50 \cdot \frac{1}{9} \cdot \frac{9}{5} \cdot \frac{1}{25} (e^{5 (x^{9})^{\frac{5}{9}}} \cdot 5 (x^{9})^{\frac{5}{9}} - e^{5 (x^{9})^{\frac{5}{9}}}) : \frac{2}{5} (5 e^{5 x^{5}} x^{5} - e^{5 x^{5}})

= \frac{2}{5} (5 e^{5 x^{5}} x^{5} - e^{5 x^{5}})

Add a constant to the solution

= \frac{2}{5} (5 e^{5 x^{5}} x^{5} - e^{5 x^{5}}) + C

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What is the integral of 50x^9e^{5x^5} ?
The integral of 50x^9e^{5x^5} is 2/5 (5e^{5x^5}x^5-e^{5x^5})+C