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Integrals

Multiple Choice Questions

501.

Let f(x) = x² - 2. If $\int_{3}^{6} f (x) dx = 3 f (c)$ for some c $\in$ (3, 6), then the value of c is equal to

$\sqrt{12}$
$\sqrt{21}$
$\sqrt{19}$
$\sqrt{17}$

502.

$\int_{0}^{\frac{π}{2}} \frac{\sin (2 x)}{1 + 2 \cos^{2} (x)} dx$ is equal to

$\frac{1}{2} \log (2)$
$\log (2)$
$\frac{1}{2} \log (3)$
$\log (3)$

503.
$\int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan^{3} (x)}$ is equal to

1

$π$

$\frac{π}{2}$

$\frac{π}{4}$

$\frac{π}{4}$

$Let I = \int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan^{3} (x)} \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\cos^{3} (x)}{\sin^{3} (x) + \cos^{3} (x)} dx . . . (i) \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\cos^{3} (\frac{π}{2} - x)}{\sin^{3} (\frac{π}{2} - x) + \cos^{3} (\frac{π}{2} - x)} dx [∵ \int_{0}^{a} f (x) = \int_{0}^{a} f (a - x) dx] \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\sin^{3} (x)}{\cos^{3} (x) + \sin^{3} (x)} On adding Eqs . (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{2}} \frac{\sin^{3} (x) + \cos^{3} (x)}{\sin^{3} (x) + \cos^{3} (x)} d x = \int_{0}^{\frac{π}{2}} (1) dx = {[x]}_{0}^{\frac{π}{2}} \Rightarrow 2 I = \frac{π}{2} \Rightarrow I = \frac{π}{4}$