Q. 43 - 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. 

    ${\int }_1^4 \frac{2x+3}{x^2+3x+4}dx$

Step-by-Step Solution

Answer

Ans:  The exact value of, ${\int }_1^4 \frac{2x+3}{x^2+3x+4}dx =\ln \left(32\right)-\ln \left(8\right)$

Step 1. Given information.

given,

       ${\int }_1^4 \frac{2x+3}{x^2+3x+4}dx$

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

The exact value is calculated as shown below, 

    $\begin{array}{r}{\int }_1^4 \frac{2x+3}{x^2+3x+4}dx\\ ={\int }_1^4 \frac{1}{u}du\\ =[\ln (\left|u\right|){]}_1^4\\ =\ln \left|4^2+3\left(4\right)+4\right|-\ln \left|(1{)}^2+3(1)+4\right|\\ =\ln (16+12+4)-\ln (1+3+4)\\ =\ln \left(32\right)-\ln \left(8\right)\end{array}$

 Therefore, the exact value is $\ln \left(32\right)-\ln \left(8\right)$

Step 3. Check:

The required graph is,