Question

For \[\int_0^{\frac\pi4}\sin^3 2t\cos 2t\,dt,\] which substitution and differential relation are correct?

Options

• \[\sin 2t=u,\qquad \cos 2t\,dt=\frac12\,du\]

• \[\sin^3 2t=u,\qquad \cos 2t\,dt=du\]

• \[\cos 2t=u,\qquad \cos 2t\,dt=2\,du\]

• \[2t=u,\qquad \cos 2t\,dt=\frac12\,du\]

Answer 1

Put \(\sin 2t=u\), so that \(2\cos 2t\,dt=du\).
Therefore \(\cos 2t\,dt=\frac12\,du\), and the integral becomes \(\frac12\int u^3\,du\).