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improper integral examples (Topic)

1. $\displaystyle\int_0^\infty e^{-x^2}\,dx \;=\; \frac{\sqrt{\pi}}{2}$

2. $\displaystyle\int_0^\infty e^{-x^2}\cos{kx}\,dx\;=\;\frac{\sqrt{\pi}}{2}e^{-\frac{1}{4}k^2}$

3. $\displaystyle\int_0^\infty \frac{e^{-x^2}}{a^2\!+\!x^2}\,dx \;=\;\frac{\pi}{2a}e^{a^2}\,{\rm erfc}\,a$

4. $\displaystyle\int_0^\infty\sin{x^2}\,dx \;=\; \int_0^\infty\cos{x^2}\,dx \;=\; \frac{\sqrt{2\pi}}{4}$

5. $\displaystyle\int_0^\infty\frac{\sin{ax}}{x}\,dx \;=\; ($sgn$\,a)\frac{\pi}{2} \qquad (a \in \mathbb{R})$

6. $\displaystyle\int_0^\infty\left(\frac{\sin{x}}{x}\right)^2 dx \;=\; \frac{\pi}{2}$

7. $\displaystyle\int_0^\infty\frac{1-\cos{kx}}{x^2}\,dx \;=\; \frac{\pi k}{2}$

8. $\displaystyle\int_0^\infty\frac{x^{-k}}{x\!+\!1}\,dx \;=\; \frac{\pi}{\sin{\pi k}} \quad (0 < k < 1)$

9. $\displaystyle\int_{-\infty}^\infty\frac{e^{kx}}{1\!+\!e^x}\,dx \;=\; \frac{\pi}{\sin{\pi k}} \quad (0 < k < 1)$

10. $\displaystyle\int_0^\infty\frac{\cos{kx}}{x^2\!+\!1}\,dx \;=\; \frac{\pi}{2e^k}$

11. $\displaystyle\int_0^\infty\frac{a\cos{x}}{x^2\!+\!a^2}\,dx \;=\; \int_0^\infty\frac{x\sin{x}}{x^2\!+\!a^2}\,dx \;=\; \frac{\pi}{2e^a} \quad\; (a > 0)$

12. $\displaystyle\int_0^\infty\frac{\sin{ax}}{x(x^2\!+\!1)}\,dx \;=\; \frac{\pi}{2}(1-e^{-a}) \quad\; (a > 0)$

13. $\displaystyle\int_0^\infty e^{-x}x^{-\frac{3}{2}}\,dx \;=\; \sqrt{\pi}$

14. $\displaystyle\int_0^\infty e^{-x}x^3\sin{x}\,dx \;=\; 0$

15. $\displaystyle\int_0^\infty\!\left(\frac{1}{e^x\!-\!1}-\frac{1}{xe^x}\right) dx \;=\; \gamma$

16. $\displaystyle\int_0^\infty\!\frac{\cos{ax^2}-\cos{ax}}{x} dx \;=\; \frac{\gamma+\ln{a}}{2} \quad (a > 0)$

17. $\displaystyle\int_0^\infty\frac{e^{-ax}\!-\!e^{-bx}}{x}\,dx \;=\; \ln\frac{b}{a} \quad (a > 0,\;\, b > 0)$

18. $\displaystyle\int_1^\infty\left(\arcsin\frac{1}{x}-\frac{1}{x}\right)\,dx \;=\; 1+\ln{2}-\frac{\pi}{2}$

19. $\displaystyle\int_0^1\frac{\arctan{x}}{x\sqrt{1\!-\!x^2}}\,dx \;=\; \frac{\pi}{2}\ln(1\!+\!\sqrt{2})$

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Other names:  list of improper integrals
Keywords:  improper integral

This is version 2 of improper integral examples, born on 2009-04-18, modified 2009-04-18.
Object id is 656, canonical name is ImproperIntegralExamples.
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Physics Classification: 02.30.-f (Function theory, analysis)
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