Evaluate the integral. \int (\cot^{3}x)dx

Stacie Worsley 2021-12-18 Answered
Evaluate the integral.
(cot3x)dx
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Expert Answer

Neil Dismukes
Answered 2021-12-19 Author has 37 answers

Step 1
Consider the given indefinite integral:
(cot3x)dx
Here the objective is to find the indefinite integral of cot3x.
Step 2
Rewrite the given indefinite integral in this form,
(cotx)×(cot2x)dx
=cotx(cosec2x1)dx Where cosec2cot2x=1
Use sum rule of integration (f(x)±g(x))dx=f(x)dx±g(x)dx
=(cotx×cosec2x)dx(cotx)dx
Let cotx=t then
ddx(cotx)dx=dt
(cosec2x)dx=dt
(cosec2x)dx=dt
Substitute cotx=tand (cosec2x)dx=dt
=(cotx×cosec2x)dxln|sinx|+c
Where (cotx)dx=ln|sinx|+c
=t(dt)ln|sinx|+c
=t22ln|sinx|+c
=cot2x2ln|sinx|+c
Hence the indefinite integral of cot3x is

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ambarakaq8
Answered 2021-12-20 Author has 31 answers
nick1337
Answered 2021-12-28 Author has 510 answers

We make a trigonometric substitution: tan(x)= and then dt=1/(1+t2)
t3t2+1dt
Simplify the expression: The
x3x2+1dx
degree of the numerator P (x) is greater than or equal to the degree of the denominator Q (x), so we divide the polynomials.
x3x2+1=x+xx2+1
By integrating the whole part, we get:
(x)dx=x22
Integrating further, we get:
xx2+1dx=ln(x2+1)2
Answer:
x22+ln(x2+1)2+C
or
x22+ln(x2+1)+C
Returning to the change of variables (t=cot(x)), we get:
I=ln(cot(x)2+1)2cot(x)22+C

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