Question

evaluate the integral.
int t ^ { 2 } left( 2 + t ^ { 3 }
ight) ^ { 4 } d t
int t ^ { 2 } left( 2 + t ^ { 3 }
ight) ^ { 4 } d t =

Explanation:

Step1: Use substitution

Let \(u = 2 + t^{3}\), then \(du=3t^{2}dt\), and \(t^{2}dt=\frac{1}{3}du\).

Step2: Change the limits of integration

When \(t = 0\), \(u=2 + 0^{3}=2\). When \(t = 1\), \(u=2+1^{3}=3\).

Step3: Rewrite the integral

\(\int_{0}^{1}t^{2}(2 + t^{3})^{4}dt=\frac{1}{3}\int_{2}^{3}u^{4}du\)

Step4: Integrate \(u^{4}\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), we have \(\frac{1}{3}\times\frac{u^{5}}{5}\big|_{2}^{3}\)

Step5: Evaluate the definite - integral

\(\frac{1}{15}(u^{5})\big|_{2}^{3}=\frac{1}{15}(3^{5}-2^{5})\)
\(=\frac{1}{15}(243 - 32)=\frac{211}{15}\)

Answer:

\(\frac{211}{15}\)