Expert: Ahmed Salami - 1/6/2006

Question
salam alikom i need the detailed solution for
tan^n(y)  integral
best regards

Answer
Hi Ali,
Sorry for the time taken.
To find the reduction formula for tan^n(x)
we know that tan^2(x) = sec^2(x) - 1
we then write as
tan^n(x) = tan^n-2(x).[sec^2(x) - 1]
The integral of tan^n(x) then becomes
$tan^n-2(x).[sec^2(x) - 1]dx
= $tan^n-2(x).sec^2(x) dx - $tan^n-2(x)dx
If I(n) = $tan^n(x)dx, then
I(n-2) = $tan^n-2(x)dx
Returning back to our integration, we'll then have
I(n) = $tan^n-2(x).sec^2(x) dx - I(n-2)
We now make use of the fact that
$[f(x)]^m . f'(x) dx = [f(x)]^(m+1) / (m + 1)
Therefore
$tan^n-2(x).sec^2(x) dx becomes tan^n-1(x) / n-1  + C
The final reduction formula becomes
I(n) = tan^n-1(x) / n-1  - I(n-2)

I hope i have helped, you can always get back to me.
Regards.