Q. 32 - 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.

  ${\int }_2^4 3e^{2x-4}dx$

Step-by-Step Solution

Answer

Ans:   The exact value of, ${\int }_2^4 3e^{2x-4}dx =\frac{3}{2}{e}^4$

Step 1. Given information.

given,

      ${\int }_2^4 3e^{2x-4}dx$

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

The exact value is calculated as shown below, 

    $\begin{array}{r}{\int }_2^4 3e^{2x-4}dx\\ =3{\int }_2^4 e^{2x-4}dx\\ =3\left(\frac{1}{2}\right){\left[e^{2x-3}\right]}_2^4\\ =\frac{3}{2}\left[e^{2\left(4\right)-3}-e^{2\left(2\right)-3}\right]\\ =\frac{3}{2}\left[e^5-e^1\right]\\ =\frac{3}{2}{e}^4\end{array}$

Therefore, the exact value is, $\frac{3}{2}{e}^4$

Step 3. Check

The required graph is,