To determine

To evaluate:

The given integral.

Answer

ln⁡|sin⁡x|+c

Explanation

1) Concept:

The substitution rule: If u=g(x) is a differentiable function whose range is I,  and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here g(x) is substituted as u and then differentiation

g’(x)dx =du

2) Formula:

i.

cot⁡x=cos⁡xsin⁡x

ii.

∫1xdx=ln⁡|x|+c

3) Given:

∫cot⁡x dx

4) Calculation:

Consider the given integral,∫cot⁡x dx

By using trigonometric identity

∫cot⁡x dx= ∫cos⁡xsin⁡x dx

Here using the substitution method

Substitute sin⁡x=u,

Differentiating with respect to x

cos⁡x dx=du,

Using this in the given integral, it becomes

∫cot⁡x dx= ∫cos⁡xsin⁡x dx= ∫1udu

By using the rule of indefinite integral

=ln⁡|u|+c

Resubstituting u=sin⁡x in the above solution,

=ln⁡|sin⁡x|+c

Conclusion:

Therefore, ∫cot⁡x dx= ln⁡|sin⁡x|+c