If the equality ∫$$0x$$ btcos⁡$$4t$$−asin⁡$$4tt2dt=asin$$⁡$$4xx$$−$$1holds$$ for all x such that $$0$$

Question

If the equality ∫$$0x$$ btcos⁡$$4t$$−asin⁡$$4tt2dt=asin$$⁡$$4xx$$−$$1holds$$ for all x such that  $$0$$

Infinity Learn · Accepted Answer

∫$$0x$$ btcos⁡$$4t$$−asin⁡$$4tt2dt=b$$∫$$0xcos4ttdt-a$$∫$$0xsin4tt2$$ $$dt=b$$∫$$0x$$ $$cos$$⁡$$4ttdt$$−a−$$sin$$⁡$$4tt0x+4$$∫$$0x$$ $$cos$$⁡$$4ttdt=(b$$−$$4a)$$∫$$0x$$ $$cos$$⁡$$4ttdt+asin$$⁡$$4xx$$−$$4aThus$$ (b$$−$$4a)∫$$0xxcos$$⁡$$4ttdt+$$ asin⁡$$4xx$$−$$4a=asin$$⁡$$4xx$$−$$1i$$.e. (b$$−$$4a)∫$$0xxcos$$⁡$$4ttdt=4a-1$$.Since R.H.S. is independent of x, so we must have b−$$4a=0$$ and $$4a$$−$$1=0$$ i.e., $$a=1/4$$,$$b=1$$.

If the equality
$\int_{0}^{x} \frac{b t \cos 4 t - a \sin 4 t}{t^{2}} d t = \frac{a \sin 4 x}{x} - 1$
holds for all x such that $0 < x < π / 4$ then a and b are given by

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If the equality ∫0x btcos⁡4t−asin⁡4tt2dt=asin⁡4xx−1holds for all x such that 0<x<π/4 then a and b are given by

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If the equality
$\int_{0}^{x} \frac{b t \cos 4 t - a \sin 4 t}{t^{2}} d t = \frac{a \sin 4 x}{x} - 1$
holds for all x such that $0 < x < π / 4$ then a and b are given by