Resolved: "Evaluate the integral. ∫cot3xtan⁡xdx"

fertilizeki

Answered question

2021-12-14

Evaluate the integral.

\int \cot^{3} x \tan x d x

William Appel

Beginner2021-12-15Added 44 answers

Step 1
Given:

\int \cot^{3} x \tan x d x

Step 2
Explanation:

\int \cot^{3} x \tan x d x = \int \frac{\cos^{3} (x)}{\sin^{3} (x)} \cdot \frac{\sin (x)}{\cos (x)} d x

= \int \frac{\cos^{2} (x)}{\sin^{2} (x)} d x

= \int \frac{1 - \sin^{2} (x)}{\sin^{2} (x)} d x

[∵ \sin^{2} x + \cos^{2} x = 1 \Rightarrow \cos^{2} x = 1 - \sin^{2} x]

= \int (\frac{1}{\sin^{2} (x)} - 1) d x

= \int \frac{1}{\sin^{2} (x)} d x - \int 1 d x

= \int \csc^{2} (x) d x - \int 1 d x

= - \cot (x) - x + C

where C being arbitrary constant
Step 3
Answer:

\int \cot^{3} (x) \tan (x) d x = - \cot (x) - x + C

Charles Benedict

Beginner2021-12-16Added 32 answers

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

\int {(\frac{1}{\tan (x)})}^{3} \cdot \tan (x) d x

\int \frac{1}{{\tan (x)}^{3}} \cdot \tan (x) d x

\int \frac{1}{{\tan (x)}^{2}} d x

\int {\cos (x)}^{2} d x

\int {\csc (x)}^{2} - 1 d x

\int {\csc (x)}^{2} d x - \int 1 d x

- \cot (x) - x

- \cot (x) - x + C

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