If a is a positive integer, then the number of values of a satisfying ∫$$0$$$$π$$$$2{a2cos3x4+34cosx+asinx-20cosx}dx$$≤$$-a23$$

Question

If a is a positive integer, then the number of values of a satisfying                                                                                                                                                                                ∫$$0$$$$π$$$$2{a2cos3x4+34cosx+asinx-20cosx}dx$$≤$$-a23$$

Infinity Learn · Accepted Answer

The L.H.S. of the above inequality is equal to $$a2sin$$⁡$$3x12+34sin$$⁡x−acos⁡x−$$20sin$$⁡$$x0$$π$$/2=a2$$−$$112+34$$−$$a(0$$−$$1)$$−$$20=2a23+a$$−$$20$$.Thus the given inequality is $$2a2/3+a$$−$$20$$≤−$$a2/3i$$.e., $$a2+a$$−$$20$$≤$$0$$⇔−$$5$$≤a≤$$4Since$$ a is a positive integer so  $$a=1$$,$$2$$,$$3$$,$$4$$.

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

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If a is a positive integer, then the number of values of a satisfying ∫0π2{a2cos3x4+34cosx+asinx-20cosx}dx≤-a23

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