What is the integral of x^3*e^{5x} ?

The integral of x^3*e^{5x} is 1/625 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C

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integral of x^3*e^{5x}

\int x^{3} \cdot e^{5 x} d x

\frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

Solution steps

\int x^{3} e^{5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{625} d u

= \int \frac{e ^{u} u ^{3}}{625} d u

Take the constant out:

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

= \frac{1}{625} \cdot \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= \frac{1}{625} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

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

= \frac{1}{625} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 5 x

= \frac{1}{625} ((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}{625} ((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}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

= \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

Add a constant to the solution

= \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

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What is the integral of x^3*e^{5x} ?
The integral of x^3*e^{5x} is 1/625 (125e^{5x}x^3-3(25e^{5x}x^2-2(5e^{5x}x-e^{5x})))+C