First slide

Evaluation of definite integrals

Question

If $f (x) = \int_{0}^{\sin^{2} x} \sin^{- 1} \sqrt{t} d t$ and $g (x) =$
$\int_{0}^{\cos^{2} x} \cos^{- 1} \sqrt{t}$ dt then the value of $f (x) + g (x)$

Moderate

A

$π$
B

$π / 4$
C

$π / 2$
D

$\sin^{2} x + \sin x + x$

Solution

$f^{'} (x) + g^{'} (x) = \sin^{- 1} (\sin x) 2 \sin x \cos x - \cos^{- 1} (\cos x) 2 \sin x \cos x$
$= x \sin 2 x - x \sin 2 x = 0$
for all $x \in R$ . Hence $f (x) + g (x) = constant = C (say)$
Putting $x = π / 4, C = \int_{0}^{1 / 2} \sin^{- 1} \sqrt{t} d t + \int_{0}^{1 / 2} \cos^{- 1} \sqrt{t} d t$
$= \int_{0}^{1 / 2} \frac{π}{2} d t = \frac{π}{4}$
Hence $f (x) + g (x) = π / 4$

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