Question

19.sin x cos3x

Answer 1

The given function is 1 sinx cos 3 x .

The given function can be written as,

1 sinx cos 3 x = sin 2 x+ cos 2 x sinx cos 3 x = sinx cos 3 x + 1 sinxcosx =tanx sec 2 x+ 1 cos 2 x sinxcosx cos 2 x =tanx sec 2 x+ sec 2 x tanx (1)

From (1), we get,

∫ 1 sinx cos 3 x dx = ∫ ( tanx sec 2 x )dx + ∫ [ 1 cos 2 x sinxcosx cos 2 x ] dx = ∫ ( tanx sec 2 x )dx + ∫ sec 2 x tanx dx (2)

Put tanx=t sec 2 xdx=dt

Substitute t for tanx and dt for sec 2 xdx in (1),

∫ 1 sinx cos 3 x dx = ∫ tdt+ ∫ 1 t dt = t 2 2 +log( t ) = tan 2 x 2 +log(tanx)+c

Thus, the integral of the function 1 sinx cos 3 x is tan 2 x 2 +log(tanx)+c.