Question

evaluate the integral.
int r ^ { 7 } left( 5 + r ^ { 8 }
ight) ^ { 7 } d r
int r ^ { 7 } left( 5 + r ^ { 8 }
ight) ^ { 7 } d r =

Explanation:

Step1: Substitute \( u = 5 + r^8 \)

Differentiate \( u \) with respect to \( r \): \( du=8r^{7}dr\), so \(r^{7}dr=\frac{1}{8}du\)

Step2: Rewrite the integral

The integral \(\int r^{7}(5 + r^{8})^{7}dr\) becomes \(\int u^{7}\cdot\frac{1}{8}du\)

Step3: Integrate with respect to \( u \)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(\frac{1}{8}\int u^{7}du=\frac{1}{8}\cdot\frac{u^{8}}{8}+C=\frac{u^{8}}{64}+C\)

Step4: Substitute back \( u = 5 + r^8 \)

The result is \(\frac{(5 + r^{8})^{8}}{64}+C\)

Answer:

\(\frac{(5 + r^{8})^{8}}{64}+C\)