Integrate 1 / (1 + cot x) using substitution method.

We will make use of simple substitution of the whole integral value.

Answer: The solution to the give expression is  (x/2) – (1/2)· log | sin x + cos x | + C

We can employ the conversion of the given expression equal to I and then solve for I.

Explanation:

Let I = ∫ 1/(1 + cot x) dx

We will solve this for I.

I =  ∫ [ 1 / (1 + (cos x / sin x)) ]dx

= ∫  [ (sin x / sin x + cos x) ]dx

= ∫ 1/2[ (2sin x / (sin x + cos x)) ]dx

=  ∫ 1/2[ {(sin x + cos x) + (sin x - cos x)} / (sin x + cos x) ]dx

= ∫ 1/2[ (sin x - cos x) / ( sin x + cos x) ]dx  + ∫ 1/2dx

= ∫ 1/2x + ∫1/2[ (sin x - cos x) / ( sin x + cos x) ]dx

Now, to solve further we will assume (sin x + cos x) = t

Or, (cos x - sin x)dx  = dt

So, I = x/2 +  1/2 ∫ -(dt)/t

= x/2 - 1/2 log | t | + C

Now, substituting back the value of t, we get 

= x/2 - 1/2 log | sin x + cos x | + C

Hence the integration of 1 / (1 + cot x) using substitution method gives the value x/2 - 1/2 log | sin x + cos x | + C.