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

The integral of x^5e^{5x} is 1/3125 e^{5x}(625x^5-625x^4+500x^3-300x^2+120x-24)+C

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integral of x^5e^{5x}

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

\frac{1}{3125} e^{5 x} (625 x^{5} - 625 x^{4} + 500 x^{3} - 300 x^{2} + 120 x - 24) + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u} u ^{5}}{15625} d u

= \int \frac{e ^{u} u ^{5}}{15625} d u

Take the constant out:

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

= \frac{1}{15625} \cdot \int e^{u} u^{5} d u

Apply Integration By Parts

: u^{5} e^{u} - \int 5 u^{4} e^{u} d u

= \frac{1}{15625} (u^{5} e^{u} - \int 5 u^{4} e^{u} d u)

\int 5 u^{4} e^{u} d u = 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))

= \frac{1}{15625} (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

Substitute back

u = 5 x

= \frac{1}{15625} ((5 x)^{5} e^{5 x} - 5 ((5 x)^{4} e^{5 x} - 4 ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x})))))

Simplify

\frac{1}{15625} ((5 x)^{5} e^{5 x} - 5 ((5 x)^{4} e^{5 x} - 4 ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x}))))) : \frac{1}{3125} e^{5 x} (625 x^{5} - 625 x^{4} + 500 x^{3} - 300 x^{2} + 120 x - 24)

= \frac{1}{3125} e^{5 x} (625 x^{5} - 625 x^{4} + 500 x^{3} - 300 x^{2} + 120 x - 24)

Add a constant to the solution

= \frac{1}{3125} e^{5 x} (625 x^{5} - 625 x^{4} + 500 x^{3} - 300 x^{2} + 120 x - 24) + C

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What is the integral of x^5e^{5x} ?
The integral of x^5e^{5x} is 1/3125 e^{5x}(625x^5-625x^4+500x^3-300x^2+120x-24)+C