evaluate the integral.\n int t ^ { 4 } left( 8 + t ^ { 5 } \right) ^ { 6 } d t \n int t ^ { 4 } left( 8 + t…

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

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

Answer

Explanation:

Step1: Use substitution

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

Step2: Change the limits of integration

When (t = 0), (u=8 + 0^{5}=8). When (t = 1), (u=8 + 1^{5}=9).

Step3: Rewrite the integral

The integral (\int_{0}^{1}t^{4}(8 + t^{5})^{6}dt) becomes (\frac{1}{5}\int_{8}^{9}u^{6}du).

Step4: Integrate (u^{6})

Using the power - rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\frac{1}{5}\times\frac{u^{7}}{7}\big|_{8}^{9}).

Step5: Evaluate the definite integral

(\frac{1}{35}(u^{7})\big|_{8}^{9}=\frac{1}{35}(9^{7}-8^{7})). Calculate (9^{7}=47239249), (8^{7}=2097152). (\frac{1}{35}(47239249 - 2097152)=\frac{45142097}{35}).

Answer:

(\frac{45142097}{35})