Q. 39 - Calculus

Question

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer. 

    ${\int }_{-3}^2 \frac{1}{x+5}dx$

Step-by-Step Solution

Answer

Ans:    The exact value is,${\int }_{-3}^2 \frac{1}{x+5}dx =\ln \left(7\right)-\ln \left(2\right)$

Step 1. Given information.

given,

      ${\int }_{-3}^2 \frac{1}{x+5}dx$

Step 2. The objective is to determine the exact value of the definite integral.

The exact value is calculated as shown below, 

      Substitute  $u=x+5$

            $$\begin{gathered} {\int }_{-3}^2 \frac{1}{x+5}dx \\ ={\int }_{-3}^2 \frac{1}{u}dx \\ = {\left[ \ln \left(u\right) \right]}_{-3}^2 \\ ={\left[ \ln (x+5) \right]}_{-3}^2 \\ =\ln \left(7\right) - \ln \left(2\right) \end{gathered}$$

Therefore, the exact value is,$\ln \left(7\right)-\ln \left(2\right)$.

Step 3. Check:

The required graph is,