Question

evaluate the integral.
int t ^ { 4 } left( 8 + t ^ { 5 }
ight) ^ { 6 } d t
int t ^ { 4 } left( 8 + t ^ { 5 }
ight) ^ { 6 } d t =

Explanation:

Step1: Substitution

Let \(u = 8 + t^{5}\), then \(du=5t^{4}dt\), and \(t^{4}dt=\frac{1}{5}du\).

Step2: Integrate with respect to \(u\)

The integral \(\int t^{4}(8 + t^{5})^{6}dt\) becomes \(\frac{1}{5}\int u^{6}du\).
Using the power - rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(\frac{1}{5}\times\frac{u^{7}}{7}+C=\frac{u^{7}}{35}+C\).

Step3: Substitute back \(u = 8 + t^{5}\)

Substituting \(u = 8 + t^{5}\) back, we get \(\frac{(8 + t^{5})^{7}}{35}+C\).

Answer:

\(\frac{(8 + t^{5})^{7}}{35}+C\)