Question

Integrate: cot(x) + cosec(x) w.r.t x

Answer 1

Answer:

$2 \ln | \sin \frac{x}{2} | + c$

Step-by-step explanation:

$\int (\cot(x) + \cosec(x) )dx\\ = \int( \frac{ \cos x}{ \sin x} + \frac{1}{ \sin \: x} )dx \\ = \int \frac{ \cos x + 1}{ \sin x} dx \\ = \int \frac{2 \cos ^{2} ( \frac{x}{2} ) }{ 2\sin( \frac{x}{2} ) \cos( \frac{x}{2} ) }dx \\ = \int \frac{ \cos( \frac{x}{2}) }{ \sin(\frac{x}{2})} dx$

Now taking

$\sin( \frac{x}{2} ) = t$

Differenting both side

$\sin( \frac{x}{2} ) = t \\ \implies \cos( \frac{x}{2} ) \times \frac{1}{2} dx = dt \\ \implies \cos( \frac{x}{2} ) dx = 2dt$

Now using the above value

$\int \frac{ \cos( \frac{x}{2} ) }{ \sin \frac{x}{2} } \\ = \int \frac{2dt}{t} \\ = 2 \ln t + c \\ = 2 \ln | \sin( \frac{x}{2} ) | + c$