Calculus: Finding Initial Velocity?

Question

Here's the problem:
What is the initial velocity that an object must be thrown upward at to reach a maximum height of 200 m (from a height of 2 m)? It says to use a(t)= -9.8 m/s/s (neglect air resistance).

The answer I got was 198 m/s, but I'm not quite sure if that's right... please help. Thanks in advance! :)
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And a quick 2nd calc problem:

antiderivative of (cos x)/(1-cos^2x)dx

It is probably really easy, but for some reason I can't figure it out... :(

And for the second one.. it is the antiderivative of (cos x)/(1-cos^(2)x)

Answer 1

Question 1
v² = u² - 2gs
0 = u² - 2 x 9.8 x 198
u² = 2 x 9.8 + 198
u² = 217.6
u = 14.8 m / s is initial velocity.

Question 2
I = ∫ (cosx / sin ² x) . dx
Let u = sinx
du = cos x dx
I = ∫(1/u²).du
I = ∫(u)^(-2) du
I = - 1 / u + C
I = - 1 / sin x + C
I = - cosec x + C

Answer 2

Puggy's answer on your integral is exactly correct, so I'll just explain the top part.

You can use the work-energy theorem and evaluate the initial velocity necessary by saying that the maximum height has velocity zero and displacement 198 meters (200 - 2)

By this we say 0 = V^2 + 2(-9.81)(198), solving for V yields the correct answer.

You can check this with the alternate forms s(t) = -1/2at^2 + Vt + 2 and v(t) = V - at making sure that at the time t where s(t) = 200 the function v(t) is zero. In other words it reaches the height 200 with a velocity of zero, this is of course the top of the path by logical induction.

Answer 3

Integral ( (cos(x)/(1 - cos^2(x)) dx )

To solve this, first convert the denominator to sin^2(x) (by the identity).

Integral ( (cos(x) / sin^2(x) dx )

We're going to use substitution. But first, I'm going to rearrange the integral to show you an intermediate step.

Integral ( 1/sin^2(x) cos(x) dx)

Use substitution.
Let u = sin(x). Then
du = cos(x) dx.
Note that the tail end of our integral is cos(x) dx, so after our substitution, the tail end will be du.

Integral ( 1/u^2 du )

Integrate directly. Note that 1/u^2 is the same as u^(-2) which, when integrated, we use the reverse power rule. Our answer is (-1)u^(-1) after integrating.

(-1)u^(-1) + C

-1/u + C

But u = sin(x), so back-substituting we get

-1/sin(x) + C

-csc(x) + C