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

The integral of x^3e^{nx} is 1/(n^4)(n^3e^{nx}x^3-3(n^2e^{nx}x^2-2(ne^{nx}x-e^{nx})))+C

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integral of x^3e^{nx}

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

\frac{1}{n ^{4}} (n^{3} e^{n x} x^{3} - 3 (n^{2} e^{n x} x^{2} - 2 (n e^{n x} x - e^{n x}))) + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{n ^{4}} d u

= \int \frac{e ^{u} u ^{3}}{n ^{4}} d u

Take the constant out:

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

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

Substitute back

u = n x

= \frac{1}{n ^{4}} ((n x)^{3} e^{n x} - 3 ((n x)^{2} e^{n x} - 2 (e^{n x} n x - e^{n x})))

Simplify

\frac{1}{n ^{4}} ((n x)^{3} e^{n x} - 3 ((n x)^{2} e^{n x} - 2 (e^{n x} n x - e^{n x}))) : \frac{1}{n ^{4}} (n^{3} e^{n x} x^{3} - 3 (n^{2} e^{n x} x^{2} - 2 (n e^{n x} x - e^{n x})))

= \frac{1}{n ^{4}} (n^{3} e^{n x} x^{3} - 3 (n^{2} e^{n x} x^{2} - 2 (n e^{n x} x - e^{n x})))

Add a constant to the solution

= \frac{1}{n ^{4}} (n^{3} e^{n x} x^{3} - 3 (n^{2} e^{n x} x^{2} - 2 (n e^{n x} x - e^{n x}))) + C

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What is the integral of x^3e^{nx} ?
The integral of x^3e^{nx} is 1/(n^4)(n^3e^{nx}x^3-3(n^2e^{nx}x^2-2(ne^{nx}x-e^{nx})))+C