Question

Evaluate integral from 0 to pi/10.
4sqrt(1+cos10x) dx
The book suggests using a half angle identity to reduce, but I can't figure out the steps.
Thanks!

Answer 1

First observe the following half-angle formula:\[\bf \cos(x)=\pm \sqrt{\frac{ 1+\cos(2x)}{2} } \implies \sqrt{2}\cos(x)=\pm \sqrt{1+\cos(2x)}\]Multiply the "x" and "2x" by 5 to get:\[\bf \sqrt{2}\cos(5x)=\pm \sqrt{1+\cos(10x)}\]So now we can make this substitution in our integral:\[\bf \int\limits_{0}^{\pi/10}4\sqrt{1+\cos(10x)} \ dx=\bf \int\limits_{0}^{\pi/10}4\sqrt{2}\cos(5x) \ dx =\left[ \frac{ 4\sqrt{2} }{ 5 }\sin(5x) \right]_{0}^{\pi/10}=?\]Can you evaluate the integral?

@bigmatt

Answer 2

@bigmatt Can you evaluate the answer?

Answer 3

Yes, I was able to. The answer is: \[\frac{ 4\sqrt{2} }{ 5 }\] I really appreciate your help! I was stuck for over an hour! Thank you!

Answer 4

Correct! Good job.

Answer 5

done! I am a fan!

Answer 6

=]

Answer 7

Good job, I really like your method of evaluation, @genius12 . The "multiplying by 5" got me for a minute to make me ask myself "why is he doing that?" Took a minute. Haha.

Answer 8

hehe lol