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##\int^{π/2}_{-π/2}\frac{4cosθ}{3+sin2θ}dθ##

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u = 3 + 2 sinθ, du = 2 cosθ dθ,

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u(-π/2)=1, u(π/2)=5

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##\int^{π/2}_{-π/2}\frac{4cosθ}{3+sin2θ}dθ=\int^{5}_{1}\frac{2}{u}du##

##=2ln|u|]^{5}_{1}##

##=2ln|5|-2ln|1|##

##=2ln5##

I wonder why -pi/2 = 1 and pi/2 = 5?

Note:

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