Evaluating improper integrals?

Question

How do i evaluate the integral of sin x / sqrt(cos x) from pi / 4 to pi /2 ? (pi / 4 on the bottom)

Answer 1

It's undefined at the endpoint of the interval (pi/2).
First go ahead and integrate it like you normally would.
integral of sin(x)/sqrt(cosx) dx
Use substitution. Let u = cos x, then du = - sinx dx
= integral of -du/ sqrt(u)
= integral -u^(-1/2) du
= -u^(-1/2 +1) / (-1/2 + 1)
= -u^(1/2) / (1/2)
= -2u^(1/2)
= -2 sqrt(u)
Replace u with cosx
= -2 sqrt(cos x)
Now plug in pi/2
- 2 sqrt(cos pi/2)
= -sqrt(0)
= 0
Plug in pi/4
-2 sqrt(cos pi/4)
= -2sqrt( sqrt(2) / 2)
Now subtract them
0 - (-2sqrt(sqrt(2) /2))
= 2sqrt( sqrt(2) / 2)

Answer 2

You are going to have to use u-substitution. Set u=cos x, which means du= -sinx dx (rewrite as -du= sinx dx). Replace what you can in the problem. You now have the integral from pi/4 to pi/2 of 1/sqrt(u) *du. Rewrite as u^-1/2, and evaluate the integral, not forgetting to multiply by the -1 on the outside. The integral before you evaluate the bounds should be -2*sqrt(cos x). Evaluate with that equation, and you should get 2^3/4.

Answer 3

I'll write that as sin x * (cos x)^-0.5.

Consider the (cos x)^-0.5 part. If we find something which differentiates to that, by the chain rule we will end up some sort of sin x multiplied to it, which is what we want.
So, how can we get a (cos x)^-0.5? That would come from 2(cos x)^0.5. When we differentiate that, we get (cos x)^-0.5 * -sin x.

Thus to get the original answer, we are only out by a factor of -1, so the original integral is -2(cos x)^0.5.

Now just substitute in the limits: -2(cos pi/2)^0.5 - -2(cos pi/4)^0.5 = 0 - -2(sqrt(2)/2)^0.5 = 1.68179, approx.

Answer 4

you can make a complex integral of it and intgerate around the pole pi/2 with an epsilon of say eps.
next you take the limit for eps -> 0

Answer 5

put cosx=t
-sinxdx=dt
sinxdx=-dt
the integral=rt t dt
=(2/3)t^3/2
(2/3[cso(pi/2)^3/2-cos(pi/4)^3/2]

Answer 6

Int r^-0.999*dr = a million/0.001 r^0.001 between (0 and a million) =1000 Int -lnx dx = (por partes) = -[x*lnx -x ) between (o and a million) lim x*ln x x=>0 = 0 (you will discover it utilising L'Hôpital to lnx/(a million/x). So at a million that's a million and at 0 that's 0 the value is1