Integration of cos square x: Formula, Proof | cos^2x Integration

The integration of cos square x is x/2+(sin2x)/4+C, C is a constant. Here we learn how to integrate cos²x. The integration formula of cos square x is given as follows:

∫cos²x dx = x/2 +(sin2x)/4 +C

where C is a constant.

How to Integrate cos^2x

First, we write down the trigonometric formula of cos²x. The formula is given below:

cos2x = 2cos²x -1.

⇒ 2cos²x = 1 + cos2x

∴ cos²x = $\dfrac{1}{2}$(1 + cos 2x)

Integrating both sides, we get

∫cos²x dx = $\dfrac{1}{2}$ ∫(1+cos 2x) dx + C where C is an arbitrary constant.

= $\dfrac{1}{2}$ [∫dx + ∫ cos 2x dx] + C

= $\dfrac{1}{2}$ [x + $\dfrac{\sin 2x}{2}$] + C as the integration of cosmx is equal to (sinmx)/m.

= $\dfrac{x}{2}$ + $\dfrac{\sin 2x}{4}$ + C.

Thus the integration of cos^2x (cosine square x) is x/2+(sin2x)/4 +C where C is an integral constant.

ALSO READ:

Integration of sin3x	Integration of tan2x
Integration of sin2x	Integration of log(sinx)

Question-Answer:

Find the definite integral of cos2x from 0 to π/2, that is, find ∫$_0^{\pi/2}$ cos2x dx

Solution:

As the integral of cos²x is equal to x/2 + (sin2x)/4 +C, we obtain that

∫$_0^{\pi/2}$ cos²x dx = [x/2 + (sin2x)/4 +C]$_0^{\pi/2}$

= [π/4 + (sinπ)/4 +C] – [0/2 + (sin0)/4+C]

= (π/4 + 0 +C) – (0 +0+C) as the value sinπ is 0 and the value of sin0 is 0.

= π/4.

So the integration of cos²x from 0 to π/2 is equal to π/4.

ALSO READ:

Integration of cot x

Integration of tan x

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