Question

integration of x square divided by under root (1+x)

Answer 1

Let x=sinθ⟹dx=cosθ dθ

∴∫1−x2√x dx=∫1−sin2θ√sinθcosθ dθ

=∫cosθsinθcosθ dθ

=∫cos2θsinθ dθ

=∫1−sin2θsinθ dθ

=∫cscθ dθ−∫sinθ dθ

=ln|cscθ−cotθ|+cosθ+C

=ln∣∣∣1x−1−x2√x∣∣∣+1−x2−−−−−√+C  

=ln∣∣∣1−1−x2√x∣∣∣+1−x2−−−−−√+C