What is the integral of x^3*e^{12x} ?

The integral of x^3*e^{12x} is 1/3456 e^{12x}(288x^3-72x^2+12x-1)+C

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integral of x^3*e^{12x}

\int x^{3} \cdot e^{12 x} d x

\frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

Solution steps

\int x^{3} e^{12 x} d x

Apply u-substitution

= \int \frac{e ^{u} u ^{3}}{20736} d u

Take the constant out:

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

= \frac{1}{20736} \cdot \int e^{u} u^{3} d u

Apply Integration By Parts

= \frac{1}{20736} (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}{20736} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 12 x

= \frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x})))

Simplify

\frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x}))) : \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

Add a constant to the solution

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

What is the integral of x^3*e^{12x} ?
The integral of x^3*e^{12x} is 1/3456 e^{12x}(288x^3-72x^2+12x-1)+C