#### To determine

**To evaluate:**

The given integral.

#### Answer

ln|sinx|+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.

cotx=cosxsinx

ii.

**∫1xdx=ln|x|+c**

**3) Given:**

∫cotx dx

**4) Calculation:**

Consider the given integral,∫cotx dx

By using trigonometric identity

∫cotx dx= ∫cosxsinx dx

Here using the substitution method

Substitute sinx=u,

Differentiating with respect to x

cosx dx=du,

Using this in the given integral, it becomes

∫cotx dx= ∫cosxsinx dx= ∫1udu

By using the rule of indefinite integral

=ln|u|+c

Resubstituting u=sinx in the above solution,

=ln|sinx|+c

**Conclusion:**

Therefore, ∫cotx dx= ln|sinx|+c