Question

please, explain me how can they solve the int
http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap14/section2/906d3/906_3.html

Answer 1

@Meepi it is from yours hihi...thanks for giving me that , quite clear

Answer 2

Yeah, and I've also checked the one you posted by making an account there, there's nowhere where they take the integral of sin^5(t) though..

\[\Large \int_{-{\pi \over 2}}^{{\pi \over 2}} 4\cos(t)\left(4\sin(t)\right)^44\,\,\, dt\]
Strip out the constants:
\[\Large4^6 \int_{-{\pi \over 2}}^{{\pi \over 2}} \cos(t)\sin^4(t)\,\,\,dt\]
let u = sin(t), du = cos(t)dt, new limits 1 and -1
\[\Large4^6 \int_{-1}^{1} u^4 du = 4^6\left[\frac{u^5}{5}\right]^{1}_{-1}\]
\[\Large 4^6\left(\frac{1}{5} - \frac{-1}{5}\right)\]
\[\Large2^{12}\cdot \frac{2}{5} = \frac{2^{13}}{5}\]

Answer 3

are you typing or there is something wrong with the net?

Answer 4

That's exactly the way I did. But compare to them, there is a little bit different. from the part I posted.

Answer 5

it is university site, right? they cannot be wrong, right? so, what is the logic?

Answer 6

They aren't taking the integral of sin^5(t),
\[\large (\frac{\sin^5(t)}{5})\bigg|^{\pi \over 2}_{-\pi \over 2}\]
is just different notation for
\[\large \left[\frac{\sin^5(t)}{5}\right]^{\pi \over 2}_{-\pi \over 2}\]

Answer 7

Not sure if that was causing the confusion

Answer 8

you see, they manipulate right on cos and sin, the limits don't change

Answer 9

Yeah, basically lets say you substitute back for u = sin(t):
\[\left[\frac{u^5}{5}\right]^1_{-1}\]
Then the limits on this thing change as well, becoming pi/2 and -pi/2 again:
\[\left[\frac{\sin^5(t)}{5}\right]^{\pi \over 2}_{-{\pi \over 2}}\]

Answer 10

ok, got it, makes sense now.

Answer 11

You can think of it as undoing the substitution, so the limit chances get undone as well

Answer 12

thanks a lot

Answer 13

I have to find out the gap between 2 ways. if not, cannot sleep

Answer 14

it is fixed. thank you very much friend

Answer 15

It's good to be that motivated though :)