evaluate the following indefinite integral.\n∫(1 + 8t^5)/(3t)dt\n∫(1 + 8t^5)/(3t)dt = \n(use parentheses to…

evaluate the following indefinite integral.\n∫(1 + 8t^5)/(3t)dt\n∫(1 + 8t^5)/(3t)dt = \n(use parentheses to clearly denote the argument of each function.)

evaluate the following indefinite integral.\n∫(1 + 8t^5)/(3t)dt\n∫(1 + 8t^5)/(3t)dt = \n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Split the fraction

$\int\frac{1 + 8t^{5}}{3t}dt=\int(\frac{1}{3t}+\frac{8t^{5}}{3t})dt=\int(\frac{1}{3t}+\frac{8}{3}t^{4})dt$

Step2: Integrate term - by - term

$\int(\frac{1}{3t}+\frac{8}{3}t^{4})dt=\frac{1}{3}\int\frac{1}{t}dt+\frac{8}{3}\int t^{4}dt$

Step3: Apply integration formulas

We know that $\int\frac{1}{t}dt=\ln|t|+C_1$ and $\int t^{n}dt=\frac{t^{n + 1}}{n+1}+C_2$ ($n\neq - 1$). So, $\frac{1}{3}\int\frac{1}{t}dt+\frac{8}{3}\int t^{4}dt=\frac{1}{3}\ln|t|+\frac{8}{3}\times\frac{t^{5}}{5}+C$

Step4: Simplify the result

$\frac{1}{3}\ln|t|+\frac{8}{15}t^{5}+C$

Answer:

$\frac{1}{3}\ln|t|+\frac{8}{15}t^{5}+C$