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Q. 37

Expert-verified
Found in: Page 373

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer. ${\int }_{-1}^{1} \frac{{2}^{x}}{{4}^{x}}dx$

Ans: The exact value of, ${\int }_{-1}^{1} \frac{{2}^{x}}{{4}^{x}}dx=\frac{3}{2\mathrm{ln}\left(2\right)}$.

See the step by step solution

## Step 1. Given information.

given,

${\int }_{-1}^{1} \frac{{2}^{x}}{{4}^{x}}dx$

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

The exact value is calculated as shown below,

$\begin{array}{r}{\int }_{-1}^{1} \frac{{2}^{x}}{{4}^{x}}dx\\ ={\int }_{-1}^{1} {2}^{x}{4}^{-x}dx\\ ={\int }_{-1}^{1} {2}^{x}{2}^{-2x}dx\\ ={\int }_{-1}^{1} {2}^{-x}dx\\ ={\left[-\frac{1}{\mathrm{ln}\left(2\right)}{2}^{-x}\right]}_{-1}^{1}\\ ={\left[-\frac{1}{{2}^{x}\mathrm{ln}\left(2\right)}\right]}_{-1}^{1}\\ =\left[-\frac{1}{{2}^{1}\mathrm{ln}\left(2\right)}+\frac{1}{{2}^{-1}\mathrm{ln}\left(2\right)}\right]\\ =\frac{3}{2\mathrm{ln}\left(2\right)}\end{array}$

Therefore, the value is $\frac{3}{2\mathrm{ln}\left(2\right)}$.

## Step 3. Check

The required graph is,