For 0 x 2, the value of _0^^2x ^-1( t),dt + _0^^2x ^-1( t),dt equals:

Question

For $0 \leq x \leq \dfrac{\pi}{2}$, the value of $\displaystyle\int_0^{\sin^2x} \sin^{-1}(\sqrt t)\,dt + \int_0^{\cos^2x} \cos^{-1}(\sqrt t)\,dt$ equals:

Accepted Answer

Let $I = f(x)$ after differentiating both terms via Leibniz rule:

$f'(x) = \sin^{-1}(\sin x)\cdot2\sin x\cos x + \cos^{-1}(\cos x)\cdot(-2\cos x\sin x)$

$= x\cdot\sin2x - x\cdot\sin 2x = 0 \Rightarrow f(x) = C$ (constant)

Evaluate at $x=\dfrac{\pi}{4}$: $I = \int_0^{1/2}\sin^{-1}\sqrt t\,dt+\int_0^{1/2}\cos^{-1}\sqrt t\,dt = \int_0^{1/2}\dfrac{\pi}{2}\,dt = \dfrac{\pi}{4}$

$	herefore f(x) = \dfrac{\pi}{4}$ for all $x$ in the domain.

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