determine the following indefinite integral.\n intleft(\frac{7}{t^{2}} - 8t^{4}\right)dt \n…

Explanation:

Step1: Split the integral

$\int\left(\frac{7}{t^{2}}-8t^{4}\right)dt=\int\frac{7}{t^{2}}dt-\int8t^{4}dt$

Step2: Rewrite the first - integral

$\int\frac{7}{t^{2}}dt = 7\int t^{- 2}dt$. Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $7\times\frac{t^{-2 + 1}}{-2 + 1}=-7t^{-1}$.

Step3: Integrate the second - integral

$\int8t^{4}dt=8\times\frac{t^{4 + 1}}{4+1}=\frac{8}{5}t^{5}$.

Step4: Combine the results

$7\int t^{-2}dt-\int8t^{4}dt=-7t^{-1}-\frac{8}{5}t^{5}+C$.

Answer:

$- \frac{7}{t}-\frac{8}{5}t^{5}+C$