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# C4 help: Integrate 2sec^2xtanx

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1. (Original post by notnek)
At FP3 level you should really be doing a question like this by recognition.

C4 students should too but some aren't taught it well so have to resort to substitution.
How would you do this by recognition?
2. (Original post by Bealzibub)
How would you do this by recognition?
You may know it as recognition/inspection/using standard patterns :

Notice that sec^2(x) is the derivative of tan(x) so consider the derivative of tan^2(x).

Alternatively notice that sec(x)tan(x) is the derivative of sec(x) so consider the derivative of sec^2(x).

If you haven't done integrals like this before then it's hard to explain here (especially without LaTeX grrr) - you have to practice it.
3. its quite straight forward
4. integral is actually tan^2(x) + c

I don't know why you replaced u with sinx / cos x
You should've integrated from that point = 2(u^2/2) + c = u^2 + c = tan^2(x) + c
5. you cant integrate tanx with respect to u. and if you substituted du with sec^2x*dx you would just end up back at the beginning with tan(x)sec^2(x) after the integral sign undoing everything you just did
6. The integral is both (tanx)^2 and (secx)^2. If you differentiate both you will see they give the same result, only thing that would change would be the constant of integration. Both answers are fine.
7. (Original post by delta-T)
Doing C4 trig, any idea how you would you integrate this? Thanks!
Integration by parts
8. (Original post by notnek)
You may know it as recognition/inspection/using standard patterns :

Notice that sec^2(x) is the derivative of tan(x) so consider the derivative of tan^2(x).

Alternatively notice that sec(x)tan(x) is the derivative of sec(x) so consider the derivative of sec^2(x).

If you haven't done integrals like this before then it's hard to explain here (especially without LaTeX grrr) - you have to practice it.
The best way to show it is:

int 2 sec^2x tanx dx = int 2 sec x d/dx(secx) dx
= int 2 sec x d(sec x), cancelling the dx
= 2 sec^2x / 2 + c = sec^2x+c.
9. (Original post by delta-T)
Ok I get that, but how would you know that if you didn't know he answer was sec^2x?
That's what I don't get
its just practice where you can spot that the integral is a function and its derivative, whenever you come across a complicated integral but is worth maybe 3 or 4 marks instead of 6 or 7, it could well be one of these ones.

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