Chapter 28, Problem 1RE

To determine

Whether the statement, “${\displaystyle \int \frac{dx}{1+2x}=\ln \left|1+2x\right|}+C$” is true or false.

Answer to Problem 1RE

The statement, “${\displaystyle \int \frac{dx}{1+2x}=\ln \left|1+2x\right|}+C$” is $\underset{\_}{\text{false}}$.

Explanation of Solution

Formula used:

Log rule for integration: ${\displaystyle \int \frac{du}{u}=\ln \left|u\right|}+C$

Calculation:

The given integral is ${\displaystyle \int \frac{dx}{1+2x}}$.

Note that $d\left(1+2x\right)=2dx$. Thus, multiply and divide the given function by 2 such as ${\displaystyle \int \frac{1}{2}\cdot \frac{2dx}{1+2x}}$. Therefore, $u=1+2x\text{ and }du=2dx$.

Then, by the log rule for integration, ${\displaystyle \int \frac{1}{2}\cdot \frac{2dx}{1+2x}}=\frac{1}{2}\ln \left|1+2x\right|+C$.

Therefore, the value of the function ${\displaystyle \int \frac{dx}{1+2x}}$ is $\frac{1}{2}\ln \left|1+2x\right|+C$. But in the given statement, the factor $\frac{1}{2}$ is excluded.

Thus, the statement, “${\displaystyle \int \frac{dx}{1+2x}=\ln \left|1+2x\right|}+C$” is $\underset{\_}{\text{false}}$.