Find the indefinite integral cos⁡ec2xcot3xdx

Question

au4gsf · Accepted Answer

Step 1
We have to find the indefinite integral:

\frac{\cos e c^{2} x}{\cot^{3} x} d x

This will be solved by substitution method since derivative of function is present in the integral.
Assuming,

t = \cot x

Differentiating,

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

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

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

Step 2
Substituting above values, we get

\int \frac{\cos e c^{2} x}{\cot^{3} x} d x = \int \frac{- d t}{t^{3}}

= - \int t^{- 3} d t

= - \frac{t^{- 3 + 1}}{- 3 + 1} + C

(since

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

)

= - \frac{t^{- 2}}{- 2} + C

= \frac{1}{2 t^{2}} + C

= \frac{1}{2 (\cot^{2} x)} + C

= \frac{1}{2} \tan^{2} x + C

(since

\tan x \cot x = 1

)
Where, C is an arbitrary constant.
Hence, value of indefinite integral is

\frac{1}{2} \tan^{2} x + C

.

Find the indefinite integral \frac{\cos ec^{2}x}{\cot^{3}x}dx