Question

What is the integral of cos cube root x??

Answer 1

I=∫cos x13dxLet x=u3⇒dx=3u2du⇒I=3∫cosu.u2duUsing integration by parts we get,I=3[u2∫cosu.du−∫{ddu(u2)∫cosu.du}du]=3[u2.sinu−∫{2u.sinudu}]=3[u2.sinu−2∫u.sinudu]Again using integration by parts we get,I=3[u2sinu+2{u.∫sinu.du−∫(ddu.u∫sinu.du)du}]=3[u2.sinu+2{−u.cosu+∫cosu.du}]=3[u2.sinu+2{−u.cosu+sinu}]+CI=3u2.sinu−6ucosu+6sinu+CAs x=u3
⇒∫cos x13dx=3x23.sinx13−6x13.cosx13+6sinx13+C