Integration question 1.

How do you integrate ∫dt/(t+7)^2(3-3t-t^2) ? Do you split the fraction into 3 and then integrate each of them with a different denominator? I'm confused.

So, I've written A/t+7 + B/(t+7)^2 + C/3-3t-t^2
1 = A(t+7)(3-3t-t^2) + B(3-3t-t^2) + C(t+5)
Let t=0, B(3-3(0)-(0)^2) = 3B B=1/2

Now I don't know which number to let t= for C?

When you let t=0, I think you should also get 1=21A+3B+5C (I think you may have wrote C out wrong too - t+7, not t+5)

To get rid of c, let t=-7, which also eliminates A, and then it looks like you're going to end up with some simultaneous eqns. Looking at your partial fractions though, I think you might have got these in the wrong order, so you might want to recheck your working.

How have I got them in the wrong order?

Is C supposed to be C(t+7)^2?