quiz 4\n1. (9 points) find the definite integral, simplifying completely. (answer may be left in terms of e…

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int_{a}^{b}u;dv=uv|{a}^{b}-\int{a}^{b}v;du$. Let $u = 16x$ and $dv=e^{2x}dx$. Then $du = 16dx$ and $v=\frac{1}{2}e^{2x}$.

Step2: Apply the integration - by - parts formula

$\int_{0}^{3}16x e^{2x}dx=\left[16x\cdot\frac{1}{2}e^{2x}\right]{0}^{3}-\int{0}^{3}\frac{1}{2}e^{2x}\cdot16dx$ $=\left[8x e^{2x}\right]{0}^{3}- 8\int{0}^{3}e^{2x}dx$

Step3: Evaluate $\left[8x e^{2x}\right]_{0}^{3}$

$\left[8x e^{2x}\right]_{0}^{3}=8\times3\times e^{2\times3}-8\times0\times e^{2\times0}=24e^{6}$

Step4: Evaluate $8\int_{0}^{3}e^{2x}dx$

Let $t = 2x$, $dt=2dx$. When $x = 0,t = 0$; when $x = 3,t = 6$. Then $8\int_{0}^{3}e^{2x}dx=4\int_{0}^{6}e^{t}dt$. $4\int_{0}^{6}e^{t}dt=4\left[e^{t}\right]_{0}^{6}=4(e^{6}-e^{0})=4(e^{6}-1)$

Step5: Calculate the final result

$\int_{0}^{3}16x e^{2x}dx=24e^{6}-4(e^{6}-1)$ $=24e^{6}-4e^{6}+4$ $=20e^{6}+4$

Answer:

$20e^{6}+4$