Basic Integration Help (Core 4)

What is a list of the integrals I need to revise? I mean if the equation wants the integral of sec5xtan5x and I know the integral of sec5x and the integral of tan5x how do I apply this to the original question? Thanks a lot mate

These questions don't require any real tricks, you just need to know your derivatives of trig functions. Some of them will probably be in your formula book in fact. If you know the derivatives of tan x, sec x, cosec x and cot x then you'll be able to do these questions. The only snag is the fact that you have 5x or 7x instead of just x, but you should know how to get past that.

Are you sure? because I am pretty sure what you are saying is wrong

Yes I'm sure. The derivative of $\sec 5x$ is $5\sec 5x \tan 5x$, for example, and the other two integrands are also constant multiples of derivatives of simple trig functions.

Integral of csc(7x)cot(7x)
1. Set u = 5x
2. Thus du = 5dx
3. Substitute:
Integral of (1/7)csc(u)cot(u)
4. Set v = csc(u)
5. Thus dv = -csc(u)cot(u)
6. Integral of (-1/7)(1dv)
7. Integral of (-1/7)(1dv) = (-1/7)v
8. Substitute v back in.
=(-1/7)csc(u) + C
9. Substitute u back in.
= (-1/7)csc(7x) + C

What I don't understand is how will the derivatives help me to find the integral? Sorry if it sounds really stupid.

I'm with you up untill Step 3, when you take out 1/7. How are you able to do that?

Thanks a lot guys! All Repped

As the derivative of sec(x) is sec(x)tan(x), the integral of sec(x)tan(x) is sec(x)?

Therefore, to refer to the original question, the integral of sec(5x)tan(5x) is sec(5x) ?

Not quite, you're out by a factor. Note that $\dfrac{d}{dx}[\sec (5x)] = 5\sec (5x)\tan (5x)$.