Let f: (-1,1) R be a continuous function.

Question

Let $f: (-1,1) 	o R$ be a continuous function. If $\displaystyle\int_0^{\sin x} f(t)\,dt = \dfrac{\sqrt3}{2}x$, then $f\left(\dfrac{\sqrt3}{2}ight)$ is equal to:

Accepted Answer

Differentiating both sides w.r.t. $x$: $f(\sin x)\cdot\cos x = \dfrac{\sqrt3}{2}$

Put $x = \dfrac{\pi}{3}$: $f\left(\sin\dfrac{\pi}{3}\right)\cos\dfrac{\pi}{3} = \dfrac{\sqrt3}{2}$

$f\left(\dfrac{\sqrt3}{2}\right)\cdot\dfrac12 = \dfrac{\sqrt3}{2} \Rightarrow f\left(\dfrac{\sqrt3}{2}\right) = \sqrt3$

Solutions · 1