# What is the Integration of cot^2x | Integral of cot^2 x

The integration of cot^2x is equal to -(cot x +x)+C. In this post, we will learn how to find the integral of cot^2x (cot square x). We will use the following two formulas:

1. ∫cosec2x dx = -cot x+C
2. ∫dx = x+C.

## Find Integration of cot2x

Answer: The integration of cot2x is given by the formula ∫cot2x dx = -(cotx +x)+C.

Explanation:

Recall the following trigonometric formula:

cosec2 x =1+cot2 x

∴ cot2 x = cosec2 x -1

Therefore, integrating both sides we obtain that

∫cot2x dx = ∫(cosec2 x-1) dx +C

⇒ ∫cot2x dx = ∫cosec2 x dx – ∫dx +C

⇒ ∫cot2x dx = -cotx – x+C by the above two formulas.

So the integration of cot^2x (cot square x) is equal to -(cotx +x)+C where C denotes an arbitrary integral constant.

## Definite Integral of cot2x

Question: Find the definite integral of cot2x from π/4 to π/2, that is,

find $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x dx$

Solution:

From the above, we have ∫cot2x dx = -cot x – x +C. Therefore, we get that

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^2 x dx$

= [-cot x -x+C]$_{\frac{\pi}{4}}^{\frac{\pi}{2}}$

= [-cot(π/2) –  π/2 +C] – [-cot(π/4) –  π/4 +C]

= -0 –  π/2 +C +1 +π/4 -C as the value of cot(π/2) is 0 and the value of cot(π/4) is 1.

= 1- π/4

= (4-π)/4.

Thus, the definite integral of cot2x from π/4 to π/2 is equal to (4-π)/4.