💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

WZ
Numerade Educator

Like

Report

Problem 50 Hard Difficulty

Prove that, for even powers of sine,
$$ \int_0^{\frac{\pi}{2}} \sin^{2n} x dx = \frac{1 \cdot 3 \cdot 5 \cdots \cdots (2n - 1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2n} \frac{\pi}{2} $$

Answer

Hence, $[(2 n+1)+1] \int_{0}^{\frac{\pi}{2}} \sin ^{2(n+1)} x d x=(2 n+1) \int_{0}^{\frac{\pi}{2}} \sin ^{2 n} x d x=(2 n+$
1) $\frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot(2 n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2 n} \cdot \frac{\pi}{2}=\frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot(2 n-1) \cdot(2(n+1)-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2 n} \frac{\pi}{2}$ . And
then we have that $\int_{0}^{\frac{\pi}{2}} \sin ^{2(n+1)} x d x=\frac{1}{2(n+1)} \frac{1 \cdot 3 \cdot 5 \cdot \ldots(2 n-1) \cdot(2(n+1)-1)}{2 n} \frac{\pi}{2}=$
$\frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot(2 n-1) \cdot(2(n+1)-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2 n \cdot 2(n+1)} \frac{\pi}{2}$
Thus the formula is true for $n+1$ and then we have that it's true for every $n$ integer with $n \geq 1 .$

Discussion

You must be signed in to discuss.

Video Transcript

It's a problem. Use truth that even powers of sign into girl From zero to pirate you sign extra Chuen is power. Yes. They haunt you. This one with this problem are the result of travel on forty nine we have on going without into Grew up from there on to pie over too fine likes to ends Power is equal once one off, then into girl from you know what? Your high cover two fine acts to unwise choose power Yes. Now we use Desiree's out to self traveler interior from zero to high over. Ugh, Fine. Act two two as power. Just replace in two in we have This is two and minus one. Okay to join hands into hero from their high over to sign two in once too. Use is a result of a king Replace in about two months to we have This is two months long over it. You and months two times to the minus three. You can once just two which reminds to into girl from zero to pry over to fine two in the final four. Yes, Then by induction we have This is the call to shut down. You come too. Two in wise. One over to in times gone Cams. Three over for hams, one over two. Passed into Girl from the Road, Pine over two and Sign two and Linus Tool and this's sign ax to zero's power. Jax. This leads adjusted Hi, Omer to this's You've got to to him, my price. One over to Intends to talk terms my half times pi over two. Is your factory this value.