The Student Room Group

Interesting problem

Show that

0π2cos2n(x)dx\int_0^{\frac{\pi }{2}} \cos ^{2 n}(x) \, dx

is

πΓ(n+12)2Γ(n+1)\frac{\sqrt{\pi }\,\Gamma \left(n+\frac{1}{2}\right)}{2 \Gamma (n+1)}
Reply 1
Wow!
:redface:
Reply 2
Original post by HapaxOromenon2
Show that

0π2cos2n(x)dx\int_0^{\frac{\pi }{2}} \cos ^{2 n}(x) \, dx

is

πΓ(n+12)2Γ(n+1)\frac{\sqrt{\pi }\,\Gamma \left(n+\frac{1}{2}\right)}{2 \Gamma (n+1)}


ETA: You might like to use the commands \displaystyle\int and \dfrac for pretty integrals and large fractions respectively.

Spoiler

(edited 8 years ago)

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