PlanetPhysics/Improper Integral Examples

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Link to the original entry from which one can find the derivations of the given values.