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Question Number 115705 by mnjuly1970 last updated on 27/Sep/20

$$...nicemath...$$$$$$$$find::$$$$$$$$$$$$\mathrm{\Phi }={\int }_0^{\mathrm{\infty }}(sin\left(x\right)-cos\left(x\right))ln\left(x\right)dx=???$$$$$$

Answered by Ar Brandon last updated on 28/Sep/20

$$\mathrm{\Phi }={\int }_0^{\mathrm{\infty }}(\sin x-\cos x)\ln x\mathrm{d}x$$$$=-{\left[(\cos x+\sin x)\ln x\right]}_0^{\mathrm{\infty }}+{\int }_0^{\mathrm{\infty }}[\frac{\sin x}{x}+\frac{\cos x}{x}]\mathrm{d}x$$$$=-{\left[(\cos x+\sin x)\ln x\right]}_0^{\mathrm{\infty }}+\frac{\pi }{2}+{\int }_0^{\mathrm{\infty }}\frac{\cos x}{x}\mathrm{d}x$$$$...$$

Commented by Olaf last updated on 28/Sep/20

$$(\cos x+\sin x)\ln x\mathrm{in}0?$$

Commented by mnjuly1970 last updated on 28/Sep/20

$$answer::=\frac{\pi }{2}-\gamma$$

Commented by Ar Brandon last updated on 28/Sep/20

I haven't yet studied that Sir. This is also because I haven't found any Ebook on this. Do you any suggestions for me, please ?��

Commented by mnjuly1970 last updated on 28/Sep/20

$$hellomyfriend$$$$book:Advancedcalculus$$$$byMurraySpiegel$$$$Tom.m.Apostol....calculus$$$$\left(1\right)$$

Commented by Ar Brandon last updated on 28/Sep/20

Thanks very much Sir.

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