The value of $$I=$$∫log⁡$$2log$$⁡$$3$$ xsin⁡$$x2sin$$⁡$$x2+$$$$sin$$⁡log⁡$$6$$−$$x2dx$$ is

Question

The value of  $$I=$$∫log⁡$$2log$$⁡$$3$$ xsin⁡$$x2sin$$⁡$$x2+$$$$sin$$⁡log⁡$$6$$−$$x2dx$$ is

Infinity Learn · Accepted Answer

Put $$x2=t$$, so $$thatI=12$$∫log⁡$$2log$$⁡$$3$$ $$sin$$⁡tsin⁡$$t+$$$$sin$$⁡(log$$⁡$$6$$−$$t)dt=12$$∫log⁡$$2log$$⁡$$3$$ $$sin$$⁡$$(log$$⁡$$2+log$$⁡$$3$$−$$t)$$sin$$⁡(log$$⁡$$2+log$$⁡$$3$$−$$t)+$$$$sin$$⁡$$tdt=12$$∫log⁡$$2log$$⁡$$3$$ $$sin$$⁡$$(log$$⁡$$6$$−$$t)$$sin$$⁡$$t+$$$$sin$$⁡(log$$⁡$$6$$−$$t)dt2I=I+I=12$$∫log⁡$$2log$$⁡$$3$$ $$sin$$⁡$$t+$$$$sin$$⁡$$(log$$⁡$$6$$−$$t)$$sin$$⁡$$t+$$$$sin$$⁡$$(log$$⁡$$6$$−$$t)dt=12$$[log⁡$$3$$−log⁡$$2$$]⇒  $$I=14log$$⁡$$32$$.

The value of $I = \int_{\sqrt{\log 2}}^{\sqrt{\log 3}} \frac{x \sin x^{2}}{\sin x^{2} + \sin (\log 6 - x^{2})} d x$ is

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The value of I=∫log⁡2log⁡3 xsin⁡x2sin⁡x2+sin⁡log⁡6−x2dx is

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The value of $I = \int_{\sqrt{\log 2}}^{\sqrt{\log 3}} \frac{x \sin x^{2}}{\sin x^{2} + \sin (\log 6 - x^{2})} d x$ is