PlanetPhysics/Improper Integral Examples

1. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}dx=\frac{\sqrt{\pi }}{2}$\\

2. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x^2}\cos kxdx=\frac{\sqrt{\pi }}{2}{e}^{-\frac{1}{4}{k}^2}$\\

3. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-x^2}}{a^2+x^2}dx=\frac{\pi }{2a}{e}^{a^2}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}a$\\

4. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\sin x^{2}dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos x^{2}dx=\frac{\sqrt{2\pi }}{4}$\\

5. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin ax}{x}dx=(\text{sgn}a)\frac{\pi }{2}(a\in ℝ)$\\

6. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{\sin x}{x}\right)}^{2}dx=\frac{\pi }{2}$\\

7. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1-\cos kx}{x^2}dx=\frac{\pi k}{2}$\\

8. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{-k}}{x+1}dx=\frac{\pi }{\sin \pi k}(0<k<1)$\\

9. \; $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{e^{kx}}{1+e^x}dx=\frac{\pi }{\sin \pi k}(0<k<1)$\\

10. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos kx}{x^2+1}dx=\frac{\pi }{2e^k}$\\

11. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{a\cos x}{x^2+a^2}dx=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x\sin x}{x^2+a^2}dx=\frac{\pi }{2e^a}(a>0)$\\

12. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin ax}{x(x^2+1)}dx=\frac{\pi }{2}(1-e^{-a})(a>0)$\\

13. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{x}^{-\frac{3}{2}}dx=\sqrt{\pi }$\\

14. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{x}^3\sin xdx=0$\\

15. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{1}{e^x-1}-\frac{1}{x{e}^x})dx=\gamma$\\

16. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos a{x}^2-\cos ax}{x}dx=\frac{\gamma +\ln a}{2}(a>0)$\\

17. \; $\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-ax}-e^{-bx}}{x}dx=\ln \frac{b}{a}(a>0,b>0)$\\

18. \; $\underset{1}{\overset{\mathrm{\infty }}{\int }}(\arcsin \frac{1}{x}-\frac{1}{x})dx=1+\ln 2-\frac{\pi }{2}$\\

19. \; $\underset{0}{\overset{1}{\int }}\frac{\arctan x}{x\sqrt{1-x^2}}dx=\frac{\pi }{2}\ln (1+\sqrt{2})$\\

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