Integral of sinx cosx sin2x cos3x dx?

Integral[sin(x)cos(x)sin2(x)cos3(x)] dx

gather terms

integral[sin3(x) cos4(x)] dx

pull one sin(x) as sin is odd

integral[sin2(x) cos4(x) sin(x)] dx

using trig identities

integral[(1 - cos2(X)) cos4(x) sin(X)] dx

u substitution

u = cos(x)

du = - sin(x) dx

so

integral[(1 - u2)) (u4) - du] dx

- integral[(1- u2)) u4 du] dx

= u - 1/3u3 + 1/5u5 du

= cos(x) - 1/3cos3(x) + 1/5cos5(x) + C

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