Question

Evaluate the integral, if it exists.

\(\int {\frac{{\cos (\ln x)}}{x}} dx\)

Short answer

The integral\(\int {\frac{{\cos (\ln x)}}{x}} dx\) exists and the value for the integral is obtained as \(I = \sin (\ln x) + c\).

Step-by-step solution

Integral definition

In mathematics, an integral is a numerical number equal to the area under a function's graph for some interval or a new function whose derivative equals the original function.

Evaluation of the integral

Evaluate the integral –

\(I = \int {\frac{{\cos (\ln x)}}{x}} dx\)

Now, consider that –

\(\begin{aligned}{c}u &= \ln x\\du &= \frac{{dx}}{x}\\dx &= xdu\end{aligned}\)

Substitute out the \(x\) and \(dx\) for \(u\) and \(du\) -

\(I = \int {(\cos u)du} \)

Substitution and further calculation

It is known that \(\int {(\cos x)dx = \sin x + c} \), so –

\(I = \sin u + c\)

Substitute back \(u = \ln x\)-

\(I = \sin (\ln x) + c\)

Therefore, the value of the integral is \(I = \sin (\ln x) + c\).