mixture: integral of { (cosx)^2 / [ (cosx)^2+ 4 (sinx)^2 ] }

Friday, January 20, 2017

integral of { (cosx)^2 / [ (cosx)^2+ 4 (sinx)^2 ] }

integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ] }

use trigonometric formulae to change (sin x)^2 = 1 - (cos x)^2 so that the intergral is completely in terms of (cos x)^2 .
Now try to write the numerator in terms of the denominator.
introduce a (-3 ) in the numerator and denominator and add and subtract 4

split it into two terms and then two integrals

The second integral contains (cos x)^2 .

divide each term with (cos x)^2 to get (sec x)^2

use trigonometric formulae tochange (sec x)^2 = 1+ (tan x)^2 in the denominator only

use substitution t = tan x and change the limits.