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September 2, 2014

September 2, 2014

Posted by **Jennifer** on Saturday, May 9, 2009 at 11:59pm.

1. Use the fundamental theorem of calculus to find the derivative:

(d/dt) the integral over [0, cos t] of (3/5-(u^2))du

I have a feeling I will be able to find the derivative easily, I'm just having trouble with the very first step-- finding the integral. The only thought I've had so far is possibly rearranging the function like this:

3(5-(u^2))^-1

but I don't know if can work like that. Any ideas?

2. Evaluate these trigonometric integrals:

integral of sin(^2)x*cosx(dx)

I think I may have to use a substitution here, but I'm not sure. I started like this:

(sin(x))^2 + (cos(x))^2 = 1

therefore (sin(x))^2=1-(cos(x))^2

so:

the integral of (1-(cos(x))^2)cosx dx

which brings us to:

integral of cosx-(cos(x))^3dx

Now I know I could split that up into two integrals, but we haven't learned the integral of (cos(X))^3. I could make a substitution where u=cosx and du/-sinx= dx, but then my integral would be:

integral of (u^3)(-cscx)du

which doesn't make the problem any more simple. Any ideas? Was it wrong to make the substitution for (sin(x))^2 at the start?

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