Question

Evaluate the integral, if it exists.

\(\int {\sin x\cos (\cos x)dx} \)

Short answer

The integral \(\int {\sin x\cos (\cos x)dx} \) exists, and the value of the integral is obtained as\( - \sin (\cos (x)) + C\).

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 –

\(\int {\sin x\cos (\cos x)dx} \)

Consider that –

\(\begin{aligned}{c}u &= \cos (x)\\du &= - \sin (x)du = x\\ - du &= \sin (x)dx\end{aligned}\)

Substituting and further calculation

Substituting \(u\) to function \(\int {\sin x\cos (\cos x)dx} \) -

\(\begin{aligned}{c}\int {\sin } (x)\cos (\cos (x))dx &= \int {\cos (u)( - du)} \\ &= - \int {\cos (u)du} \\ &= - \sin (u) + C\end{aligned}\)

Reverting back to the value of \(u\), it can be seen –

\(\begin{aligned}{c}\int {\sin } (x)\cos (\cos (x))dx &= - \sin (u) + C\\ &= - \sin (\cos (x)) + C\end{aligned}\)

Therefore, the value of integral is\( - \sin (\cos (x)) + C\).