I couldn't wait for a solution!!

Calculus Level 3

Compute the integral

$\displaystyle\int_0^\pi$ $\sqrt{(1+cos2x)/2}$ $dx$

I know the solution.............. I was not getting any name for this

$dx$ is square root free

1 solution

Brian Charlesworth
Jan 12, 2015

As $\cos(2x) = 2\cos^{2}(x) - 1$ we have that $\sqrt{\dfrac{1 + \cos(2x)}{2}} = |\cos{x}|$ .

So by symmetry the integral becomes

$2\displaystyle\int_{0}^{\frac{\pi}{2}} \cos(x) dx = 2(\sin(\frac{\pi}{2}) - \sin(0)) = \boxed{2}$ .

Easy as pie ! Nice solution sir ! :D

Keshav Tiwari - 6 years, 5 months ago

Mental calculus for me. Definitely over-rated.

Sharky Kesa - 6 years, 4 months ago

You're quite right. And the answer would be the same if it were $\sin(2x)$ rather than $\cos(2x)$ .

Brian Charlesworth - 6 years, 4 months ago