Verified answer

∫cos⁵x*sinxdx=-∫t⁵dt=-t⁶/6=-cos⁶x/6+C

cosx=t   -sinxdx=dt   sinxdx=-dt

∫x³∛(x-1)dx=3∫(t³+1)³*t*t²dt=∫(t¹²+3t⁹+3t⁶+t³)dt=3(t¹³/13+3t¹⁰/10+3t⁷/⁷+t⁴/4)=3((∛x-1)¹³/13+3(∛x-1)¹⁰/10+3(∛x-1)^7/7+(∛x-1)^⁴/4+C

x-1=t³   x=t³+1   dx=3t²dt