compute the following integral using integration by parts. use c to represent an constant. ∫4x³eˣ dx

Explanation:

Step1: Recall integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 4x^{3}$ and $dv=e^{x}dx$. Then $du = 12x^{2}dx$ and $v = e^{x}$.

Step2: Apply integration - by - parts

$\int 4x^{3}e^{x}dx=4x^{3}e^{x}-\int 12x^{2}e^{x}dx$.

Step3: Apply integration - by - parts again

For $\int 12x^{2}e^{x}dx$, let $u = 12x^{2}$, $dv=e^{x}dx$. Then $du = 24x dx$ and $v = e^{x}$. So $\int 12x^{2}e^{x}dx=12x^{2}e^{x}-\int 24x e^{x}dx$.

Step4: Apply integration - by - parts again

For $\int 24x e^{x}dx$, let $u = 24x$, $dv=e^{x}dx$. Then $du = 24dx$ and $v = e^{x}$. So $\int 24x e^{x}dx=24x e^{x}-\int 24e^{x}dx$.

Step5: Integrate the last integral

$\int 24e^{x}dx = 24e^{x}+C$.

Step6: Back - substitute

$\int 4x^{3}e^{x}dx=4x^{3}e^{x}-(12x^{2}e^{x}-(24x e^{x}-24e^{x}))+C$. Simplify the expression: [ \begin{align*} \int 4x^{3}e^{x}dx&=4x^{3}e^{x}-12x^{2}e^{x}+24x e^{x}-24e^{x}+C\ &=e^{x}(4x^{3}-12x^{2}+24x - 24)+C \end{align*} ]

Answer:

$e^{x}(4x^{3}-12x^{2}+24x - 24)+C$