evaluate the definite integral below. \n\n\\(\\int_{3}^{5}(8\\sqrt{8x + 6})dx\\)

Explanation:

Step1: Use substitution

Let $u = 8x+6$, then $du=8dx$. When $x = 3$, $u=8\times3 + 6=30$. When $x = 5$, $u=8\times5+6 = 46$.

Step2: Rewrite the integral

The integral $\int_{3}^{5}(8\sqrt{8x + 6})dx=\int_{30}^{46}\sqrt{u}du$.

Step3: Integrate $\sqrt{u}$

The antiderivative of $\sqrt{u}=u^{\frac{1}{2}}$ is $\frac{2}{3}u^{\frac{3}{2}}$ according to the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$).

Step4: Evaluate the definite integral

$\left[\frac{2}{3}u^{\frac{3}{2}}\right]_{30}^{46}=\frac{2}{3}(46^{\frac{3}{2}}-30^{\frac{3}{2}})$. $46^{\frac{3}{2}}=\sqrt{46^{3}}=\sqrt{97336}=312$. $30^{\frac{3}{2}}=\sqrt{30^{3}}=\sqrt{27000}\approx164.32$. $\frac{2}{3}(46^{\frac{3}{2}}-30^{\frac{3}{2}})=\frac{2}{3}(312 - 164.32)=\frac{2}{3}\times147.68\approx98.45$.

Answer:

$98.45$