A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup:

G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions of the Cambridge Philosophical Society 6 (1838) 379--402] that the intensity of the light along a normal line to the caustic is proportional to the square of the integral $$ \int_0^{\infty } \cos \frac{\pi}{2} \left(w^3-m w\right) \; dw $$ where $m$ is proportional to the displacement from the caustic toward the convex side. This led to the Airy functions and the Airy differential equation. The modern definition is $$ \mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt $$ and the differential equation is $y''=x\,y$. A second independent solution to the differential equation $Bi(x)$ also has an integral representation, but it is not so nice.

As a side note, if your calculus or differential equations courses teach students how to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have an integral representation) or as in this case, multiply every third, Airy's example shows a natural application of such formulas. The Airy functions are linear combinations of the series $$ y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+% \frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots $$ $$ y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2% \cdot 5\cdot 8}{10!}x^{10}+\cdots $$

Remark: While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are "complete" integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies. The gamma function is also of the first class (i.e. "complete"). The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.

A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup:

G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions of the Cambridge Philosophical Society 6 (1838) 379--402] that the intensity of the light along a normal line to the caustic is proportional to the square of the integral $$ \int_0^{\infty } \cos \frac{\pi}{2} \left(w^3-m w\right) \; dw $$ where $m$ is proportional to the displacement from the caustic toward the convex side. This led to the Airy functions and the Airy differential equation. The modern definition is $$ \mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt $$ and the differential equation is $y''=x\,y$. A second independent solution to the differential equation $Bi(x)$ also has an integral representation, but it is not so nice.

As a side note, if your calculus or differential equations courses teach students how to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have an integral representation) or as in this case, multiply every third, Airy's example shows a natural application of such formulas. The Airy functions are linear combinations of the series $$ y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+% \frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots $$ $$ y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2% \cdot 5\cdot 8}{10!}x^{10}+\cdots $$

Remark: While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are "complete" integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies. The gamma function is also of the first class (i.e. "complete"). The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.

A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup:

G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions of the Cambridge Philosophical Society 6 (1838) 379--402] that the intensity of the light along a normal curve to the caustic is proportional to the square of the integral $$ \int_0^{\infty } \cos \frac{\pi}{2} \left(w^3-m w\right) \; dw $$ where $m$ is proportional to the displacement from the caustic toward the convex side. This led to the Airy functions and the Airy differential equation. The modern definition is $$ \mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt $$ and the differential equation is $y''=x\,y$. A second independent solution to the differential equation $Bi(x)$ also has an integral representation, but it is not so nice.

As a side note, if your calculus or differential equations courses teach students how to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have an integral representation) or as in this case, multiply every third, the Airy's example shows a natural application of such formulas. The Airy functions are linear combinations of the series $$ y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+% \frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots $$ $$ y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2% \cdot 5\cdot 8}{10!}x^{10}+\cdots $$

Remark: While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are "complete" integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies. The gamma function is also of the first class (i.e. "complete"). The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.

A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup:

G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions of the Cambridge Philosophical Society 6 (1838) 379--402] that the intensity of the light along a normal line to the caustic is proportional to the square of the integral $$ \int_0^{\infty } \cos \frac{\pi}{2} \left(w^3-m w\right) \; dw $$ where $m$ is proportional to the displacement from the caustic toward the convex side. This led to the Airy functions and the Airy differential equation. The modern definition is $$ \mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt $$ and the differential equation is $y''=x\,y$. A second independent solution to the differential equation $Bi(x)$ also has an integral representation, but it is not so nice.

As a side note, if your calculus or differential equations courses teach students how to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have an integral representation) or as in this case, multiply every third, Airy's example shows a natural application of such formulas. The Airy functions are linear combinations of the series $$ y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+% \frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots $$ $$ y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2% \cdot 5\cdot 8}{10!}x^{10}+\cdots $$

Remark: While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are "complete" integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies. The gamma function is also of the first class (i.e. "complete"). The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.