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

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

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

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

2 e^{5 t^{5}} t^{5} - \frac{2}{5} e^{5 t^{5}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 50 \cdot \int \frac{e ^{u} u}{125} d u

Take the constant out:

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

= 50 \cdot \frac{1}{125} \cdot \int e^{u} u d u

Apply Integration By Parts

= 50 \cdot \frac{1}{125} (e^{u} u - \int e^{u} d u)

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

= 50 \cdot \frac{1}{125} (e^{u} u - e^{u})

Substitute back

u = 5 t^{5}

= 50 \cdot \frac{1}{125} (e^{5 t^{5}} \cdot 5 t^{5} - e^{5 t^{5}})

Simplify

50 \cdot \frac{1}{125} (e^{5 t^{5}} \cdot 5 t^{5} - e^{5 t^{5}}) : 2 e^{5 t^{5}} t^{5} - \frac{2}{5} e^{5 t^{5}}

= 2 e^{5 t^{5}} t^{5} - \frac{2}{5} e^{5 t^{5}}

Add a constant to the solution

= 2 e^{5 t^{5}} t^{5} - \frac{2}{5} e^{5 t^{5}} + C

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