0-sin-1-x-1-pi-sin-pi-x-dx-

Question Number 133563 by bemath last updated on 23/Feb/21

\underset{0}{\int^{\infty}} [\sin \frac{1}{x} - \frac{1}{π} \sin (\frac{π}{x})] dx

Answered by EDWIN88 last updated on 23/Feb/21

Calculate \underset{0}{\int^{\infty}} (\sin (\frac{1}{x}) - \frac{1}{π} \sin (\frac{π}{x})) dx

let \frac{1}{x} = t \land x = \frac{1}{t}; \begin{array}{cc} x \to \infty & t \to 0^{+} \\ x \to 0^{+} & t \to \infty \end{array}

I = \underset{\infty}{\int^{0}} (\sin t - \frac{1}{π} \sin π t) (- \frac{1}{t^{2}}) dt

I = \underset{0}{\int^{\infty}} (\frac{\frac{\sin t}{t} - \frac{\sin π t}{π t}}{t}) dt; Frullani integral

I = [\lim_{t \to \infty} (\frac{\sin t}{t}) - \lim_{t \to 0} \frac{\sin π t}{π t}] . \ln (\frac{b}{a}); {\begin{cases} b = 1 \\ a = π \end{cases}

I = (0 - 1) \ln (\frac{1}{π}) = \ln π

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