To determine

To evaluate:

The indefiniteintegral ∫sin⁡xxdx

Answer

-2cos⁡x+C

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du.

ii) Indefinite integral

∫sin⁡x dx=-cos⁡x+C

2) Given:

∫sin⁡xxdx

3) Calculation:

Here, use the substitution method because the differential of the function x is 12xdx

Substitute u=x.

Differentiate u=x with respect to x.

du=12xdx

As dxx is a part of the integration, solve for dxx by multiplying both side by 2.

2du=dxx

By using concept i),

substitute u=x ,  dxx=2du to get

∫sin⁡xxdx=∫sin⁡u2du

=2∫sin⁡udu By using concept ii),

=2(-cos⁡u)+C

Resubstitute u=x.

=2(-cos⁡x)+C

=-2cos⁡x+C

Conclusion:

Therefore,

∫sin⁡xxdx=-2cos⁡x+C