1. ∫ ₀^∞e^−x2 dx =

2. ∫ ₀^∞e^−x2 cos kxdx = e^−k2

3. ∫ ₀^∞ dx = e^a2 erfca

4. ∫ ₀^∞ sin x² dx = ∫ ₀^∞ cos x² dx =

5. ∫ ₀^∞dx = (sgn a) (a ∈ ℝ)

6. ∫ ₀^∞2dx =

7. ∫ ₀^∞ dx =

8. ∫ ₀^∞ dx = (0 < k < 1)

9. ∫ _−∞^∞ dx = (0 < k < 1)

10. ∫ ₀^∞ dx =

11. ∫ ₀^∞ dx = ∫ ₀^∞ dx = (a > 0)

12. ∫ ₀^∞ dx = (1 − e^−a) (a > 0)

13. ∫ ₀^∞e^−xx⁻ dx =

14. ∫ ₀^∞e^−xx³ sin xdx = 0

15. ∫ ₀^∞dx = γ

16. ∫ ₀^∞dx = (a > 0)

17. ∫ ₀^∞dx = ln (a > 0, b > 0)

18. ∫ ₁^∞ dx = 1 + ln 2 −

19. ∫ ₀¹ dx = ln(1+)

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