1. evaluate the following integrals. a) ∫(3t⁴ + 2)⁵t³dt

Explanation:

Step1: Use substitution

Let $u = 3t^{4}+2$, then $du=12t^{3}dt$, and $t^{3}dt=\frac{1}{12}du$.

Step2: Rewrite the integral

The integral $\int(3t^{4}+2)^{5}t^{3}dt$ becomes $\frac{1}{12}\int u^{5}du$.

Step3: Integrate with power - rule

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\frac{1}{12}\times\frac{u^{6}}{6}+C=\frac{u^{6}}{72}+C$.

Step4: Substitute back

Substitute $u = 3t^{4}+2$ back into the result, we get $\frac{(3t^{4}+2)^{6}}{72}+C$.

Answer:

$\frac{(3t^{4}+2)^{6}}{72}+C$