evaluate the integral below. no calculator - give your answer in exact form.\n int_{10}^{17} \frac{(ln…

Explanation:

Step1: Use substitution

Let $u = \ln x$. Then $du=\frac{1}{x}dx$. When $x = 10$, $u=\ln10$; when $x = 17$, $u=\ln17$. The integral $\int_{10}^{17}\frac{(\ln x)^{3}}{x}dx$ becomes $\int_{\ln10}^{\ln17}u^{3}du$.

Step2: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int_{\ln10}^{\ln17}u^{3}du=\left[\frac{u^{4}}{4}\right]_{\ln10}^{\ln17}$.

Step3: Evaluate the definite integral

$\frac{(\ln17)^{4}}{4}-\frac{(\ln10)^{4}}{4}=\frac{(\ln17)^{4}-(\ln10)^{4}}{4}$.

Answer:

$\frac{(\ln17)^{4}-(\ln10)^{4}}{4}$