First slide

Evaluation of definite integrals

Question

If $a$ is a positive integer, then the number of values of a satisfying $\int_{0}^{\frac{π}{2}} {a^{2} (\frac{\cos (3 x)}{4} + \frac{3}{4} \cos (x)) + a \sin (x) - 20 \cos (x)} d x \leq - \frac{a^{2}}{3}$

Moderate

A

one
B

two
C

three
D

four

Solution

The L.H.S. of the above inequality is equal to
$a^{2} (\frac{\sin 3 x}{12} + \frac{3}{4} \sin x) - a \cos x - {20 \sin x|}_{0}^{π / 2} =$
$a^{2} (- \frac{1}{12} + \frac{3}{4}) - a (0 - 1) - 20 = \frac{2 a^{2}}{3} + a - 20 .$
Thus the given inequality is $(2 a^{2} / 3) + a - 20 \leq - a^{2} / 3$
i.e., $a^{2} + a - 20 \leq 0 \Leftrightarrow - 5 \leq a \leq 4$
Since $a$ is a positive integer so $a = 1, 2, 3, 4$ .

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