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

Question

Evaluate the following definite integral:

\(\int\limits_0^{\pi/2}sin\, \text x\,sin\,2\text x\,d\text x \)

Answer 1

\(\int\limits_0^{\pi/2}sin\, \text x\,\times sin\,2\text x\,d\text x \)

= \(\int\limits_0^{\pi/2}sin\, \text x\,\times2\times sin\,\text x\,cos\,\text x\,d\text x \)

⇒ \(\int\limits_0^{\pi/2}sin\, \text x\,\times sin\,(2\text x)\,d\text x \) = 2\(\int\limits_0^{\pi/2} \)(1 - cos² x) cos x dx

First let us find,

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

⇒ \(\int\limits_0^{\pi/2} \) sin x \(\times\)sin(2x) dx = \(\Big(\cfrac23\Big)\)