I have two questions, because I'm preparing for a math test on monday.
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?
4 answers