Integration of cos square x

Cos square x integration

What is Integration of cos square x dx ? In this article we will derive the integral of cosine squared x – Solving the Integral of cos²x. ∫cos²x dx = ? So Directly we can not just integrate cos²x. We have to convert cos²x into another form by using trigonometric identities to find the integration.

This integration of cos² x dx cannot be evaluated by the directly by using any formula or integration formula, so we have to use trigonometric identity of half angle cos square x or cos² x = (1+cos2x) / 2.

• I = ∫cos²x dx
• I =∫{(1+cos2x)/2} dx
• ⇒ I = (1/2)∫(1+cos2x) dx
• ⇒ I = (1/2) ∫1dx + (1/2) ∫cos2x dx
• Using the integral formula ∫cos kx dx = (sin kx) / k+c, we have
  ∫cos²x dx = (1/2) x + (1/2) (sin2x) / 2 + c
• ⇒ ∫cos²x dx = (1/2) x +(1/4) (sin2x) + c, where C is integration constant

Integration of cos square x by using double angle formula

The double angle trigonometric identity formula: cos 2x = cos² x – sin² x. There is one more trigonometric identity which will be very useful to find integration of cos square x. Second trigonometric identity is sin² x + cos² x = 1. Now we have to combine these both trigonometric identities cos 2x = cos² x – sin² x and sin² x + cos² x = 1. After combining we will get cos 2x = 2cos² x -1.

• cos 2x = 2cos² x -1
• After rearranging cos² x = ( 1+ cos 2x ) / 2, Now we can use reverse chain rule to integrate this expression.
• We will get  x/2 + ( sin 2x ) / 4 + c, where C is a integration constant.

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