Chapter 4: Q. 58 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{1} \frac{2^{x} - 3^{x}}{4^{x}} d x$

Short Answer

Expert verified

The value of integral is $(\frac{1}{2} (\frac{1}{\ln (\frac{1}{2})}) - \frac{1}{\ln (\frac{1}{2})}) - (\frac{3}{4} (\frac{1}{\ln (\frac{3}{4})}) - \frac{1}{\ln (\frac{3}{4})})$

And the graph is

Step by step solution

Step 1. Given information

An expression is given as $\int_{0}^{1} \frac{2^{x} - 3^{x}}{4^{x}} d x$

Step 2. Evaluating integral

Simplify the integral as,

$\int_{0}^{1} \frac{2^{x} - 3^{x}}{4^{x}} d x = \int_{0}^{1} {(\frac{2}{4})}^{x} d x - \int_{0}^{1} {(\frac{3}{4})}^{x} d x = \int_{0}^{1} {(\frac{1}{2})}^{x} d x - \int_{0}^{1} {(\frac{3}{4})}^{x} d x = {[\frac{1}{\ln (\frac{1}{2})} {(\frac{1}{2})}^{x}]}_{0}^{1} - {[\frac{1}{\ln (\frac{3}{4})} {(\frac{3}{4})}^{x}]}_{0}^{1} = (\frac{1}{2} (\frac{1}{\ln (\frac{1}{2})}) - \frac{1}{\ln (\frac{1}{2})}) - (\frac{3}{4} (\frac{1}{\ln (\frac{3}{4})}) - \frac{1}{\ln (\frac{3}{4})})$

And the graph is

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Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{1} \frac{2^{x} - 3^{x}}{4^{x}} d x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating integral

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Use the Fundamental Theorem of Calculus to find the exactvalues of each of the definite integrals in Exercises 19–64. Usea graph to check your answer.∫012x-3x4xdx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Probability and Statistics

Geometry

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{1} \frac{2^{x} - 3^{x}}{4^{x}} d x$