Evaluate the following definite integral: ∫ cos x cos 2x dx, x ∈[0, π/6]

Question

Evaluate the following definite integral:

\(\int\limits_0^{\pi/6} \)cos x cos 2x dx

Answer 1

 \(\int\limits_0^{\pi/6} \)cos x \(\times\)cos (2x) dx =  \(\int\limits_0^{\pi/6} \)cos x \(\times\)(cos²x - 1) dx

⇒ \(\int\limits_0^{\pi/6} \)cos x \(\times\)cos (2x) dx =  \(\int\limits_0^{\pi/6} \)(2cos³ x -cos x dx)

⇒  \(\int\limits_0^{\pi/6} \)cos x \(\times\)cos (2x) dx = 2\(\int\limits_0^{\pi/6} \)cos³x dx - \(\int\limits_0^{\pi/6} \) cos x dx

We know,

Let sin x = t. Hence, cos x dx = dt. For second expression,