question evaluate the integral below. ∫₂⁴ xe⁴ˣ dx be sure to place the argument of any trigonometric or…

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int_{a}^{b}u\mathrm{d}v=uv|{a}^{b}-\int{a}^{b}v\mathrm{d}u$. Let $u = x$ and $\mathrm{d}v=e^{4x}\mathrm{d}x$. Then $\mathrm{d}u=\mathrm{d}x$ and $v=\frac{1}{4}e^{4x}$.

Step2: Calculate $uv|{a}^{b}-\int{a}^{b}v\mathrm{d}u$

[ \begin{align*} \int_{2}^{4}xe^{4x}\mathrm{d}x&=\left[\frac{1}{4}xe^{4x}\right]{2}^{4}-\int{2}^{4}\frac{1}{4}e^{4x}\mathrm{d}x\ &=\frac{1}{4}(4e^{16}-2e^{8})-\frac{1}{16}\left[e^{4x}\right]_{2}^{4}\ &=e^{16}-\frac{1}{2}e^{8}-\frac{1}{16}(e^{16}-e^{8}) \end{align*} ]

Step3: Simplify the expression

[ \begin{align*} e^{16}-\frac{1}{2}e^{8}-\frac{1}{16}e^{16}+\frac{1}{16}e^{8}&=(1 - \frac{1}{16})e^{16}+(-\frac{1}{2}+\frac{1}{16})e^{8}\ &=\frac{15}{16}e^{16}-\frac{7}{16}e^{8} \end{align*} ]

Answer:

$\frac{15}{16}e^{16}-\frac{7}{16}e^{8}$