Question: "Integrate ∫0π1+cos⁡(2x)2dxIt is obvious that by using trig..."

znacimavjo

Answered question

2022-04-21

Integrate

\int_{0}^{π} \sqrt{\frac{1 + \cos (2 x)}{2}} d x

It is obvious that by using trig identities we can come up with:

\int_{0}^{π} \sqrt{\cos^{2} (x)} =

= \int_{0}^{π} | \cos (x) | d x

utloverej

Beginner2022-04-22Added 15 answers

| \cos (x) |

means absolute value of

\cos (x)

| \cos x | = (\cos (x),, if \cos (x) \geq 0), (- \cos,, if \cos (x) < 0) :

\cos (x) \geq 0

when x is from 0 to

\frac{π}{2}

and

\cos x \leq 0

when x is from

\frac{π}{2}

π

\int_{0}^{π} | \cos (x) | d x = \int_{0}^{\frac{π}{2}} | \cos (x) | d x + \int_{\frac{π}{2}}^{π} | \cos (x) | d x

= \int_{0}^{\frac{π}{2}} \cos (x) d x + \int_{\frac{π}{2}}^{π} - \cos (x) d x

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