evaluate the integral of (y^2)/( (1-4y^2)^0.5) with respect to y with lower limit 0 and upper limit 0.5?

Question

By means of the substitution y= k cos x, where k is a positive constant to be determined, evaluate the integral of (y^2)/( (1-4y^2)^0.5) with respect to y with lower limit 0 and upper limit 0.5. Give your answer in exact form.

Answer 1

By means of the substitution y= k cos x, where k is a positive constant to be determined, evaluate the ∫(y²/√(1-4y²)) dy with lower limit 0 and upper limit 0.5. Give your answer in exact form.

Let y = ½cosx. Therefore dy = -½sinx dx
and √(1-4y²) = √(1-4(½cosx)²) = sinx

Also when y = 0, x = π/2 and when y = 1/2, x = 0

So ∫[from y = 0 to 0.5) (y²/√(1-4y²)) dy
= ∫[from x = π/2 to 0) (1/4 cos²x /sinx * -½sinx dx
= 1/8∫[from x = 0 to π/2] cos²x dx
= 1/8∫[from x = 0 to π/2] 1/2 (cos2x + 1)dx
= 1/16∫[from x = 0 to π/2 ) [1/2 sin2x + x] [from x = 0 to π/2]
= 1/16 (0 + π/2]
= π/32

Answer 2

Which k do you think will be most useful? We need to take the square root of (1-4y^2), so if we make k =0.5, that will turn into 1-(cos x)^2 = (sin x)^2, and we can take the square root.
So, let y = 0.5 cos x.
Then dy/dx = -0.5 sin x, so dy = -0.5 sin x dx.
The y^2 on the top becomes 0.25(cos x)^2.
Finally, what are the limits of the integral? Since we took the positive square root before, we want to make sure sin x is >= 0. So y=0 when cos x = 0, which gives x = pi/2. y = 0.5 when cos x = 1, so x = 0.
So we have the integral from pi/2 to 0 of 0.25(cos x)^2/sin x * -0.5sin x dx
= integral from pi/2 to 0 of -0.125(cos x)^2 dx.
To do that, we turn (cos x)^2 into (cos 2x + 1)/2.
So we want to integrate -1/16 (cos 2x + 1).
That becomes -1/16 (1/2 sin 2x + x).
Substituting in the two limits gives 0 - -pi/32 = pi/32.
So its pi/32.
I'm 99% sure I made a mistake somewhere, but you get the idea.

Answer 3

f(x) = ?(0 to pi, (one million + cos(x)) dx ) First, evaluate the fundamental. The fundamental of one million is x, and the fundamental of cos(x) is sin(x). Defining F(x) to be the fundamental, F(x) = x + sin(x), with the intention to evaluate our sure fundamental: F(pi) - F(0) [ pi - sin(pi) ] - [ 0 - sin(0) ] [ pi - 0 ] - [ 0 - 0 ] pi - 0 pi

Answer 4

After much thought and calculation the answer is James Blunt

Answer 5

0.098175, which is pi/32