for the integral $int_{2}^{8}3xsqrt{x + 7}dx$, if we use the substitution $u=x + 7$, we can rewrite the…

for the integral $int_{2}^{8}3xsqrt{x + 7}dx$, if we use the substitution $u=x + 7$, we can rewrite the integral in terms of the variable $u$.

for the integral $int_{2}^{8}3xsqrt{x + 7}dx$, if we use the substitution $u=x + 7$, we can rewrite the integral in terms of the variable $u$.

Answer

Explanation:

Step1: Find $x$ and $dx$ in terms of $u$

Given $u = x + 7$, then $x=u - 7$ and $dx=du$.

Step2: Change the limits of integration

When $x = 2$, $u=2 + 7=9$. When $x = 8$, $u=8 + 7 = 15$.

Step3: Substitute into the integral

Substitute $x = u - 7$ and $dx=du$ into the integral $\int_{2}^{8}3x\sqrt{x + 7}dx$. We get $\int_{9}^{15}3(u - 7)\sqrt{u}du=\int_{9}^{15}(3u\sqrt{u}-21\sqrt{u})du=\int_{9}^{15}(3u^{\frac{3}{2}}-21u^{\frac{1}{2}})du$.

Answer:

$\int_{9}^{15}(3u^{\frac{3}{2}}-21u^{\frac{1}{2}})du$