Tips on Integrating Trig Functions

Date: 02/01/2001 at 00:47:35
From: Karan
Subject: Integration 

Dear Math,

I've been trying to integrate this funtion for quite some time now:

     dx/(3+4sin x)^2

I've tried to use u = 1/(3+4sinx), but the answer doesn't come out 
right.

When I saw the solution, it was solved in a way that required you to 
practically know the answer beforehand. They used y = cosx/(3+4sinx), 
differentiated it, separated the desired integral and integrated the 
rest for the answer.

Is there any other way? Why isn't my method working?

Thanks,
Karan

Date: 02/01/2001 at 11:28:12
From: Doctor Rob
Subject: Re: Integration 

Thanks for writing to Ask Dr. Math, Karan.

Such tricks are clever, but hard to discover.

There is another way that works whenever you have a rational function 
of trigonometric functions, as you do here. The rational function is 
1/(3+4*u)^2 and the trigonometric function is u = sin(x).

The systematic method is to use the substitution:

     z = tan(x/2).

Then

     sin(x) = 2*z/(1+z^2)
     cos(x) = (1-z^2)/(1+z^2)
         dx = 2*dz/(1+z^2)

Substitute this in, and you'll end up needing to integrate a rational 
function of z:

       INTEGRAL dx/(3+4*sin[x])^2
     = INTEGRAL 1/(3+4*2*z/[1+z^2])^2 * 2*dz/(1+z^2)
     = INTEGRAL (1+z^2)^2/(3*z^2+8*z+3)^2 * 2*dz/(1+z^2)
     = INTEGRAL 2*(1+z^2)/(3*z^2+8*z+3)^2 * dz

Now you can split this using the fact that

     3*z^2 + 8*z + 3 = 3*(z + [4+sqrt(7)]/3)*(z + [4-sqrt(7)]/3)

and partial fractions to break it into parts which you can easily 
integrate.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/24/2001 at 14:09:11
From: Karan
Subject: Integration

Please integrate:

   1. ((tanx)^2+2)^0.5

   2. (1-x^2)^n from (0,1) where n is an integer. Do I have to get a 
      reduction formula? Is there on other way?

Thanks,
Karan

Date: 02/26/2001 at 09:23:42
From: Doctor Rob
Subject: Re: Integration

Thanks for writing back, Karan.

1. Use the very non-obvious substitution

        tan(x) = sqrt(2)*sinh(y)

to reduce this to the integral of 1 + sech(2*y) dy. This may be a bit 
easier to deal with.

2. You can avoid the reduction formulas by using the Binomial Theorem 
to expand (1-x^2)^n into a sum, and then integrating term by term. I 
don't think you can get the integrated sum into a simple closed form 
for general n.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/