Question

Integration question help with step by step explanation. Will reward medal and fan. *Question attached below*

Answer 1

I would let cos^2x = (1 + cos2x)/2

and then I would do normal integration.

Answer 2

f(x) = sin^2 (x)
f(a-x) = sin^2 (90 -x) pay attention at the limit of the integral, it says a = pi/2 =90
f(a-x) = (sin (90 -x))^2 =cos^2 (x)
put all under the integral
the proof done

Answer 3

I'm still kinda lost

Answer 4

\[\int_0^\color{red}{a} sin^2(x)dx=\int_0^\color{red}{a}f(\color{red}{a}-x)dx\]
that is what they want you to apply to the problem
\[\int_0^\color{red}{\pi/2}sin^2(x)dx =\int_0^\color{red}{\pi/2}sin^2(\color{red}{\pi/2}-x)dx\]
however, \(sin^2(\pi/2 -x)= cos^2 (x)\)
so that, it becomes \[\int_0^\color{red}{\pi/2}cos^2(x) dx\]
Proof done

Answer 5

is it clear?

Answer 6

Very clear! Thanks so much :D

Answer 7

How do I show that the answer equals to pi/4?

Answer 8

@Loser66

Answer 9

Use the trig identity: cos(2x) = 2cos^2(x) - 1 to integrate.

Answer 10

cos^2(x) = 1/2 * (cos(2x) + 1)

Answer 11

Huh? @ranga

Answer 12

\[\int\limits_0^{\pi/2}\cos^2(x) dx = \frac{ 1 }{ 2 }\int\limits_0^{\pi/2}(\cos(2x) + 1) dx = ?\]

Answer 13

I wish I knew how to do integration
But thank you!

Answer 14

Well you need to go over the notes/text books so you will be able to integrate on your own. If I just give the answers it won't help you to learn.