Question

integral of cot(theta) d theta​

Answer 1

$\implies \displaystyle \int \: { \sf cot( \theta) d\theta}$

We know that :-

$\boxed{ \bf \: cot( x) = \frac{ \: cos \: x}{sin \: x} }$

$\implies \displaystyle \int \: { \sf \frac{ \: cos( \theta) }{sin ( \theta) } d\theta}$

We can solve the above problem using substitution method.

$\: { \sf let \: \: sin\theta } = t$

Now differentiating both sides :-

$\: { \sf \: \: cos\theta \: d\theta } = dt$

$\implies \displaystyle \int \: { \sf \frac{ \: cos( \theta) \: d\theta}{sin ( \theta) } }$

$\implies \displaystyle \int \: { \sf \frac{ dt}{t} }$

we know that :-

$\underline{\boxed{\bf{ \displaystyle \int \frac{1}{a} \: da = \sf log(a) + c }}}$

$\implies \displaystyle \int \: { \sf \frac{ dt}{t} } = \sf \: log(t) + c$

${ \sf put \: \: t = sin\theta}$

$\implies \sf \: log(sin \theta) + c$

Answer :

$\displaystyle \int \: { \sf cot( \theta) d\theta} = \sf \: log(sin \theta) + c$

Where c is constant.

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Learn more :-

∫ 1 dx = x + C

∫ sin x dx = – cos x + C

∫ cos x dx = sin x + C

∫ sec2 dx = tan x + C

∫ csc2 dx = -cot x + C

∫ sec x (tan x) dx = sec x + C

∫ csc x ( cot x) dx = – csc x + C

∫ (1/x) dx = ln |x| + C

∫ ex dx = ex+ C

∫ ax dx = (ax/ln a) + C