Question

$\large{\sf{Question :}}$
Evaluate $\sf\int\dfrac{\cot\:x}{\sin\:x}dx$​

Answer 1

Question :

Evaluate

$\sf\int\dfrac{\cot\:x}{\sin\:x}dx$

Answer :

We have to evaluate :

$\sf\int\dfrac{\cot\:x}{\sin\:x}dx$

$\sf\int\dfrac{\cos\:x}{\sin^2x}dx$

Let $\sf\:sin\:x=t$

Now, Differentiate it with respect to x

$\sf\implies\dfrac{dt}{dx}=\cos\:x$

$\sf\implies\:dt=\cos\:x\:dx$

Then,

$\sf\int\dfrac{\cos\:x}{\sin^2x}dx$

$\sf=\int\dfrac{\cos\:x}{t^2}\times\dfrac{dt}{\cos\:x}$

$\sf=\int\dfrac{dt}{t^2}$

$\sf=\int\:t^{-2}dt$

We know that

$\sf\int\:x^n=\dfrac{x^{n+1}}{n+1}$

Then,

$\sf=\dfrac{t^{-2+1}}{-2+1}+c$

$\sf=-t^{-1}+c$

$\sf=\dfrac{-1}{t}+c$

$\sf=\dfrac{-1}{\sin\:x}+c$

It is the required solution..

Answer 2

• $\tt\int\dfrac{\cot\:x}{\sin\:x}dx$

• $\tt\int\dfrac{\cos\:x}{\sin^2x}dx$

Let sin x = t

Now, Differentiate it with respect to x

• $\rm \implies\dfrac{dt}{dx}=\cos\:x$

• $\rm \implies\:dt=\cos\:x\:dx$

• $\tt→\int\dfrac{\cos\:x}{\sin^2x}dx$

• $\tt→\int\dfrac{\cos\:x}{t^2}\times\dfrac{dt}{\cos\:x}$

• $\tt→\int\dfrac{dt}{t^2}$

• $\tt→\int\:t^{-2}dt$

We know that

• $\sf\int\:x^n=\dfrac{x^{n+1}}{n+1}$

Then,

• $\tt→\dfrac{t^{-2+1}}{-2+1}+c$

• $\tt→-t^{-1}+c$

• $\tt→\dfrac{-1}{t}+c$

• $\tt→\dfrac{-1}{\sin\:x}+c$