Question

Integrate the function: $\frac {1}{1+cot \ x}$

Answer 1

$Let\ I = ∫\frac {1}{1+cot \ x} dx$

$I = ∫\frac {1}{1+\frac {cos\ x}{sin \ x}} dx$

$I = ∫\frac {sin\ x}{sin\ x + cos \ x}\ dx$

$I = \frac 12∫\frac {2sin\ x}{sin\ x + cos \ x}\ dx$

$I = \frac 12 ∫\frac {(sin \ x + cos \ x)+(sin\ x - cos\ x)}{(sin \ x + cos \ x)}dx$

$I = \frac 12∫1dx+\frac 12∫\frac {sin\ x - cos\ x}{/sin\ x + cos\ x} dx$

$I = \frac 12(x)+\frac 12∫\frac {sin\ x - cos\ x}{/sin\ x + cos\ x} dx$

$Let\ sin \ x + cos\ x = t ⇒ (cos \ x- sin\ x) dx = dt$

$∴ I = \frac x2+\frac 12 ∫\frac {-(dt)}{t}$

$I = \frac x2-\frac 12\ log \ |t|+ C$

$I = \frac x2-\frac 12\ log \ |sin \ x + cos\ x|+ C$