How to integrate ?

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Answer 1 · 2015-12-23T23:28:29

#intx^2/sqrt(4-x^2)dx#

Let's #x = 2sin(u)#

#u = arcsin(1/2x)#

#dx = 2cos(u)du#

#int4sin^2(u)/(2cos(u))*2cos(u)du# (Just magical)

#int 4sin^2(u)du#

#2int1-cos(2u)du#

#[2u-2sin(2u)]+C#

Substitute back

#2[arcsin(1/2x)-sin(2arcsin(1/2x))]+C#

which is #[2arcsin(x/2)-1/2 x sqrt(4-x^2)]#