Q. 24 - Calculus

Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

$\int \left(\frac{2}{3x}+1\right)dx$

Step-by-Step Solution

Answer

The answer is $x+\frac{2}{3}\ln \left(\right|x\left|\right)+C$.

Step 1. Given Information.

The given integral is $\int \left(\frac{2}{3x}+1\right)dx$.

Step 2. Integration.

On integrating,

$$\begin{gathered} \int \left(\frac{2}{3x}+1\right)dx \\ =\int dx+\int \frac{2}{3x}dx \\ =x+\frac{2}{3}\ln \left(\right|x\left|\right)+C \end{gathered}$$

Step 3. Verification.

On differentiating $x+\frac{2}{3}\ln \left(\right|x\left|\right)+C,$ we get,

$$\begin{gathered} 1+\frac{2}{3}\times \frac{1}{x} \\ =\left(\frac{2}{3x}+1\right) \end{gathered}$$

Hence proved.