First slide

Evaluation of definite integrals

Question

If $\int_{\sin x}^{1} t^{2} f (t) d t = 1 - \sin x,$ then $f (1 / \sqrt{3})$ is equal to

Moderate

A

1/3
B

$1 / \sqrt{3}$
C

$\sqrt{3}$
D

3

Solution

Differentiating both the sides, we obtain
$\Rightarrow f (\sin x) = \frac{1}{\sin^{2} x},$ if $\cos x \neq 0$
$\Rightarrow f (\sin x) = \frac{1}{\sin^{2} x},$ if $\cos x \neq 0$
If $\sin x = 1 / \sqrt{3}$ then $\cos x \neq 0$ so $f (1 / \sqrt{3}) = 3 .$

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