Which identity? You will need to leave the final answer in terms of x.Solving the following integral:
So my qs is
put tan(theta/2) in terms of x where x=tan(theta)
Could derive it, but imagine me sitting in the 4U trials thinking
This identity isn't expected to be memorised for the HSC, but you should be able to derive it if needed.
Well we didn't need that formula to do that integral in the original post. We could avoid half-angles as I showed. But if you did introduce half-angles, the way to get out of them is to use the half-angle formulas in some way, because these formulas tell us how to go from sin(a/2) and cos(a/2) (and hence tan(a/2)) to trig functions of a. (That's how I derived that formula above.)Could derive it, but imagine me sitting in the 4U trials thinking
x=tan(theta)
t=tan(theta/2)
"Oh boy better derive that weird squareroot cosine formula!!"
haha so is there any clue in the question or solution that allows me to conclude that formula is needed
Seem to be gettingWhich identity? You will need to leave the fnal answer in terms of x.
For that integral, you could also use a trig. substitution of x = a*tan(theta).
You end up with something involving integral of cosec(theta), which is:
-ln(cosec(theta) + cot(theta)).
Then you could convert this back to in terms of x, using trig. identities or a right-angled triangle to assist you. This avoids use of tan(theta/2).
Seem to be getting
Ohh ok - Will make sure to remember that
Pretty sure you can just back-differentiate to obtain the answer.Solving the following integral:
I subbed x=tan(theta)
t=atan(theta/2)
Ohh ok - Will make sure to remember that
Teacher told me its not a standard integral, so its best to use t-formula
If you dont mind, can you show me how I can finish off the solution from
Given t=tan(theta/2)
(just curious thats all)
Thankyou once again Integrand
Ohh ok - Will make sure to remember that
Teacher told me its not a standard integral, so its best to use t-formula
If you dont mind, can you show me how I can finish off the solution from
Given t=tan(theta/2)
(just curious thats all)
I personally think this situation is simply not necessary.
Refer to my previous post.T-formulae does allow the integral of cosec to come out nice and tidy, but there is no need for it at all
It's only tidy, if you have cosec by itself. (And it's still not necessary either, but it makes the final answer tidy)Refer to my previous post.
Also half the time a trig substitution is unnecessary for indefinite integrals.
But for definite integrals they are very slick.