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Found in: Page 306

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# Evaluate the indefinite integral$$\int {\frac{{dx}}{{5 - 3x}}}$$.

The indefinite integral value of the given equation is$$\int {\frac{{dx}}{{5 - 3x}}} = - \frac{1}{3}\ln (5 - 3x) + c$$.

See the step by step solution

## Step 1 Definition of the indefinite integral

An integral that has no upper bound and no lower bound is known as indefinite integral.

## Step 2: Expand the number of variables in the equation.

Given:$$I = \int {\frac{{dx}}{{5 - 3x}}}$$

Let $$u = 5 - 3x$$

\begin{aligned}{c}du &= - 3dx\\dx &= - \frac{{du}}{3}\end{aligned}

Substitute$$5 - 3x$$and $$- 3dx$$for $${\rm{u}}$$ and $${\rm{du}}$$

$$I = - \frac{1}{3}\int {\frac{1}{u}}$$

## Step 3: Evaluate the equation.

Integrate

$$I = - \frac{1}{3}\ln u + c$$

Substitute $$u = 5 - 3x$$

$$I = - \frac{1}{3}\ln (5 - 3x) + c$$

Therefore, the indefinite integral value of the given equation is$$\int {\frac{{dx}}{{5 - 3x}}} = - \frac{1}{3}\ln (5 - 3x) + c$$.