Can we analyze the integral.∫(cot3x)&nbsp;dx&nbsp;

Question

Can we analyze the integral.∫(cot3x)&amp;nbsp;dx&amp;nbsp;

Neil Dismukes · Accepted Answer

Step 1
Consider the given indefinite integral:

\int (\cot^{3} x) d x

Here the objective is to find the indefinite integral of

\cot^{3} x

.
Step 2
Rewrite the given indefinite integral in this form,

\int (\cot x) \times (\cot^{2} x) d x

= \int \cot x (\cos e c^{2} x - 1) d x

Where

\cos e c^{2} - \cot^{2} x = 1

Use sum rule of integration

\int (f (x) \pm g (x)) d x = \int f (x) d x \pm \int g (x) d x

= \int (\cot x \times \cos e c^{2} x) d x - \int (\cot x) d x

Let

\cot x = t

then

\frac{d}{d x} (\cot x) d x = d t

(- \cos e c^{2} x) d x = d t

(\cos e c^{2} x) d x = - d t

Substitute

\cot x = t

and

(\cos e c^{2} x) d x = - d t

= \int (\cot x \times \cos e c^{2} x) d x - \ln | \sin x | + c

Where

\int (\cot x) d x = \ln | \sin x | + c

= \int t (- d t) - \ln | \sin x | + c

= - \frac{t^{2}}{2} - \ln | \sin x | + c

= - \frac{\cot^{2} x}{2} - \ln | \sin x | + c

Hence the indefinite integral of

\cot^{3} x

is

ambarakaq8 · Accepted Answer

It is required to calculate:  ∫(cot3x)dx  Lets

nick1337 · Accepted Answer

We make a trigonometric substitution: tan⁡(x)= and then dt=1/(1+t2) ∫−t3t2+1∗dt Simplify the expression: The ∫−x3x2+1∗dx 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+1∗dx=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)2−cot⁡(x)22+C

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

Answered question

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