Integrating cos^2(x)+5sin^2(x)

Firstly, note that cos^2(x)+5sin^2(x)= cos^2(x) +sin^2(x) +4sin^2(x).

By trignoemtric identies, cos^2(x)+sin^2(x)=1 and so we can just integrate 1+4sin^2(x) since this is equal to cos^2(x)+5sin^2(x).

Again, by trignometric identities, 4sin^2(x)=4(1/2-1/2 cos(2x))=2-2cos(2x),

and so 1+4sin^2(x)=3-2cos(2x).

We can now integrate this much more easily...

3 integrates to 3x and -2cos(2x) integrates to -sin(2x).

Hence the integral, remembering the constant of integration, is...

3x -sin(2x) +c