Q. 40 - 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.)

$\int \frac{\cos \left(\ln x\right)}{x}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{\cos \left(\mathrm{lnx}\right)}{\mathrm{x}}\mathrm{dx}=\sin \left(\mathrm{In}x\right)+\mathrm{C}$.

Step 1. Given Information

Solving the given integrals. 

$\int \frac{\cos \left(\ln x\right)}{x}\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

Let

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

Step 3. This substitution changes the integral into

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