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We use integrals of left coideal subalgebras to develop Harmonic analysis forsemisimple Hopf algebras. We show how $N^*,$ the space of functional on $N,$ isembedded in $H^*.$ We define a bilinear form on $N^*$ and show that irreducible$N$-characters are orthogonal with respect to that form. We then give anexplicit formula for induced characters of $N$ and show how the inducedcharacters are embedded in $R(H).$  In the second part we give an intrinsic definition for solvable semisimpleHopf algebras via left coideal subalgebras and their integrals. We show howthis definition generalizes solvability for finite groups. In particular,commutative and nilpotent Hopf algebras are solvable. We finally prove ananalogue of Burnside theorem: A semisimple quasitriangular Hopf algebras ofdimension $p^aq^b$ is solvable.

Solvability for semisimple Hopf algebras via integrals