∫（0→a)x^2*(√(a^2-x^2))dx,

Question

∫（0→a)x^2*(√(a^2-x^2))dx,

Answer 1

令x=asinu,√(a²-x²)=acosu,dx=acosudu,u：0--->π/2
∫ [0---->a] x²√(a²-x²) dx
=∫ [0---->π/2] a²sin²u*acosu*acosu du
=a⁴∫ [0---->π/2] sin²ucos²u du
=a⁴/4∫ [0---->π/2] 4sin²ucos²u du
=a⁴/4∫ [0---->π/2] sin²2u du
=a⁴/8∫ [0---->π/2] (1-cos4u) du
=a⁴/8(u-1/4sin4u) |[0---->π/2]
=a⁴π/16