Challenging Integrals Every Mathematician Should Know

1. Dirichlet Integral

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

Technique: Feynman's trick (differentiation under the integral sign) or contour integration.

2. Gaussian Integral

$$\int_{-\infty}^{\infty} e^{-x^2} \, dx \;=\; \sqrt{\pi}$$

Technique: Polar coordinate trick — square the integral and convert to 2D.

3. Cauchy Distribution Normalization

$$\int_{-\infty}^{\infty} \frac{1}{1+x^2} \, dx \;=\; \pi$$

Technique: Direct antiderivative via $\arctan$, or the residue theorem.

4. Fresnel Integrals

$$\int_0^{\infty} \sin(x^2) \, dx \;=\; \int_0^{\infty} \cos(x^2) \, dx \;=\; \frac{1}{2}\sqrt{\frac{\pi}{2}}$$

Technique: Contour integration over a wedge-shaped contour in $\mathbb{C}$.

5. Euler's Log-Sine Integral

$$\int_0^{\pi/2} \ln(\sin x) \, dx \;=\; -\frac{\pi}{2} \ln 2$$

Technique: Symmetry and the reflection formula for $\Gamma$.

6. Basel-Type Log Integral

$$\int_0^{1} \frac{\ln x}{1-x} \, dx \;=\; -\frac{\pi^2}{6}$$

Technique: Expand $\frac{1}{1-x}$ as a geometric series and integrate term by term.

7. Log-Gamma Integral

$$\int_0^{1} \ln \Gamma(x) \, dx \;=\; \frac{1}{2}\ln(2\pi)$$

Technique: Kummer's Fourier series for $\ln\Gamma(x)$.

8. Parametric Log Integral

$$\int_0^{1} x^a \ln x \, dx \;=\; -\frac{1}{(a+1)^2}, \qquad a > -1$$

Technique: Differentiate $\int_0^1 x^a \, dx = \frac{1}{a+1}$ with respect to $a$.

9. Beta Function

$$\int_0^{1} x^{\,p-1}(1-x)^{q-1} \, dx \;=\; \frac{\Gamma(p)\,\Gamma(q)}{\Gamma(p+q)}, \qquad p,q > 0$$

Technique: Relate to the Gamma function via substitution and convolution.

10. Gamma Function Definition

$$\int_0^{\infty} x^{\,s-1} e^{-x} \, dx \;=\; \Gamma(s), \qquad s > 0$$

Technique: Foundational definition; the functional equation $\Gamma(s+1) = s\,\Gamma(s)$ follows by parts.

11. Ramanujan's Master Theorem (special case)

$$\int_0^{\infty} \frac{x^{\,s-1}}{1+x} \, dx \;=\; \frac{\pi}{\sin(\pi s)}, \qquad 0 < s < 1$$

Technique: Residue theorem or the Beta identity $B(s,\,1-s) = \frac{\pi}{\sin(\pi s)}$.

12. Wallis Integral

$$\int_0^{\pi/2} \sin^n x \, dx \;=\; \frac{\sqrt{\pi}}{2} \cdot \frac{\,\Gamma\!\left(\dfrac{n+1}{2}\right)}{\Gamma\!\left(\dfrac{n}{2}+1\right)}$$

Technique: Reduction formula or Beta function.

13. Frullani's Integral

$$\int_0^{\infty} \frac{f(ax) - f(bx)}{x} \, dx \;=\; \bigl[f(0) - f(\infty)\bigr]\ln\!\frac{b}{a}$$

Technique: Swap order of integration using $\int_a^b f'(tx)\,dt$.

14. Ahmed's Integral

$$\int_0^{1} \frac{\arctan\!\left(\sqrt{x^2+2}\right)}{(x^2+1)\sqrt{x^2+2}} \, dx \;=\; \frac{5\pi^2}{96}$$

Technique: Differentiation under the integral sign with a two-parameter family.

15. Sophomore's Dream

$$\int_0^{1} x^{-x} \, dx \;=\; \sum_{n=1}^{\infty} n^{-n}$$ $$\int_0^{1} x^{x} \, dx \;=\; -\!\sum_{n=1}^{\infty} (-n)^{-n}$$

Technique: Expand $x^{-x} = e^{-x\ln x}$ as a power series and integrate term by term.

16. Double Integral for $\zeta(2)$

$$\int_0^{1}\!\int_0^{1} \frac{1}{1-xy} \, dx \, dy \;=\; \frac{\pi^2}{6}$$

Technique: Geometric series expansion and Euler's Basel result.

17. Parseval–Plancherel Identity (sinc)

$$\int_{-\infty}^{\infty} \left(\frac{\sin x}{x}\right)^{\!2} dx \;=\; \pi$$

Technique: Parseval's theorem applied to the Fourier transform of the sinc function.

18. Power-Type Rational Integral

$$\int_0^{\infty} \frac{x^{\,p-1}}{1+x^n} \, dx \;=\; \frac{\pi}{n\,\sin\!\left(\dfrac{p\pi}{n}\right)}, \qquad 0 < p < n$$

Technique: Keyhole contour with a branch cut along the positive real axis.

19. Rational Function over $\mathbb{R}$

$$\int_{-\infty}^{\infty} \frac{x^2}{(x^2+1)(x^2+4)} \, dx \;=\; \frac{\pi}{3}$$

Technique: Partial fractions or upper half-plane residues.

20. Poisson Integral

$$\int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos\theta + r^2} \, d\theta \;=\; 2\pi, \qquad |r| < 1$$

Technique: Real part of a geometric series in $re^{i\theta}$.

21. Gaussian with Cosine Modulation

$$\int_{-\infty}^{\infty} e^{-x^2} \cos(2bx) \, dx \;=\; \sqrt{\pi}\, e^{-b^2}$$

Technique: Complete the square in the exponent; reduces to the Gaussian integral.

22. Glasser's Master Theorem

$$\int_{-\infty}^{\infty} f\!\left(x - \frac{1}{x}\right) dx \;=\; \int_{-\infty}^{\infty} f(x) \, dx$$

Technique: The substitution $x \mapsto x - 1/x$ is measure-preserving on $\mathbb{R}$.

23. Laplace Transform of $\sqrt{t}$

$$\int_0^{\infty} \sqrt{t}\; e^{-st} \, dt \;=\; \frac{\sqrt{\pi}}{2\, s^{3/2}}, \qquad s > 0$$

Technique: Substitute $u = st$ and recognize $\Gamma\!\left(\tfrac{3}{2}\right) = \tfrac{\sqrt{\pi}}{2}$.

24. Dirichlet Kernel Integral

$$\int_0^{\pi} \frac{\sin\!\left(\left(n+\tfrac{1}{2}\right)x\right)}{\sin(x/2)} \, dx \;=\; \pi$$

Technique: Telescoping and orthogonality of trigonometric functions.

25. Mellin–Barnes Type

$$\int_0^{\infty} \frac{\ln x}{1 + x^2} \, dx \;=\; 0$$

Technique: Split at $x=1$, substitute $x \mapsto 1/x$ on one piece — the two halves cancel.