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Basically, what I did was I considered the sphere as a sum of infinitesimally thin spherical shells.

The moment of inertia for one shell is dI=(r^2)*dm

where dm=(M/V)*4*pi*r^2*dr

where V=(4/3)*pi*R^3

so the equation dI=3*pi*M*r^4*dr when simplified.

Integrating this from 0 to R (Summing up the spherical shells from the center to the edge of the big sphere) gives me (3/5)*M*R^2. What is wrong with this derivation? :(