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