Q. 58 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{1}{x\ln x}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{1}{x\ln x}\mathrm{d}x=\ln \left(\mathrm{lnx}\right)+\mathrm{C}$.

Step 1. Given Information

Solving the given integrals.

$\int \frac{1}{x\ln x}\mathrm{d}x$

Step 2. Using the substitution method.

Let

$$\begin{gathered} u=\ln x \\ \frac{du}{dx}=\frac{1}{x} \\ du=\frac{1}{x}dx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int \frac{1}{x\ln x}\mathrm{d}x=\int \frac{1}{u}\mathrm{du} \\ \int \frac{1}{\mathrm{xlnx}}\mathrm{dx}=\mathrm{lnu}+\mathrm{C} \\ \int \frac{1}{\mathrm{xlnx}}\mathrm{dx}=\ln \left(\mathrm{lnx}\right)+\mathrm{C} \end{gathered}$$