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Problem 45 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{6}} \sqrt{1 + \cos 2x} dx $


$$\frac{1}{2} \sqrt{2}$$


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Video Transcript

this problem is from Chapter seven section to a problem. Number forty five in the book Calculus Early Transcendental lt's a Condition by James Store. We hear we have a definite integral of the square of one plus coast high to X. So here let's talk about co sign of two X. We can use a double ankle formula to rewrite this as two times co sign squared X minus one. Then from here we have one plus coastline to X. Here we can cancel the ones and we just have to co sign Squared X. Then we could take the square root so we can write. This is I want to times the square room of Coastline Square, which isn't necessarily co sign because technically, the square root of the square is going to be the absolute value. So this case, we should to be safe. You should always write absolute value there, I guess in general, the rule that I'm using here is squirt of a square is absolute value in our problem. However, this will be actually will simplify too, two times called radical two times co sign since coastline eggs. It's positive in our inseparable of the liberation. What that said we can rewrite are integral. That becomes zero time for six time's radical, too times cosine x so you can pull out the rat radical to outside the integral if you like. So Integral co sign a sign so they get radical, too time. Sinus evaluated at zero and pi over six. So sign of viruses. One half and sign of zero zero So we get squared or two divided by two, and that's our answer.