evaluate the definite integral below. ∫₃⁶ -7/x² dx

Explanation:

Step1: Rewrite the integrand

Rewrite $\frac{-7}{x^{2}}$ as $-7x^{- 2}$. So the integral becomes $\int_{3}^{6}-7x^{-2}dx$.

Step2: Apply the 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 $\int - 7x^{-2}dx=-7\int x^{-2}dx$. Using the power - rule, we get $-7\times\frac{x^{-2 + 1}}{-2+1}=-7\times\frac{x^{-1}}{-1}= \frac{7}{x}+C$.

Step3: Evaluate the definite integral

Use the fundamental theorem of calculus $\int_{a}^{b}f(x)dx=F(b)-F(a)$, where $F(x)$ is an antiderivative of $f(x)$. Here, $F(x)=\frac{7}{x}$, $a = 3$, and $b = 6$. So $F(6)-F(3)=\frac{7}{6}-\frac{7}{3}$.

Step4: Simplify the result

$\frac{7}{6}-\frac{7}{3}=\frac{7 - 14}{6}=-\frac{7}{6}$.

Answer:

$-\frac{7}{6}$