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Q. 34

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. ${\int }_{2}^{4} \frac{1}{4-3x}dx$

Ans: The exact value of, $\phantom{\rule{0ex}{0ex}}{\int }_{2}^{4} \frac{1}{4-3x}dx=-\frac{1}{3}\left(\mathrm{ln}8\right)+\frac{1}{3}\left(\mathrm{ln}2\right)$

See the step by step solution

## Step 1. Given information.

given,

${\int }_{2}^{4} \frac{1}{4-3x}dx$

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

The exact value is calculated as shown below,

$\begin{array}{r}{\int }_{2}^{4} \frac{1}{4-3x}dx\\ =-\frac{1}{3}\left[\mathrm{ln}\left(|4-3x|\right){\right]}_{2}^{4}\\ =-\frac{1}{3}\left[\mathrm{ln}\left(|4-3\left(4\right)|\right)-\mathrm{ln}\left(|4-3\left(2\right)|\right)\right]\\ =-\frac{1}{3}\left[\mathrm{ln}\left(8\right)-\mathrm{ln}\left(2\right)\right]\\ =-\frac{1}{3}\left(\mathrm{ln}8\right)+\frac{1}{3}\left(\mathrm{ln}2\right)\end{array}$

Therefore, the exact value is localid="1648629053226" $-\frac{1}{3}\left(\mathrm{ln}8\right)+\frac{1}{3}\left(\mathrm{ln}2\right)$

## Step 3. Check

The required graph is,