Many challenging integration problems can be solved surprisingly quickly by simply knowing the right technique to apply. While finding the right technique can be a matter of ingenuity, there are a dozen or so techniques that permit a more comprehensive approach to solving definite integrals.

Manipulations of definite integrals may rely upon specific limits for the integral, like with odd and even functions, or they may require directly changing the integrand itself, through some type of substitution. However, most integrals require a combination of techniques, and many of the more complicated approaches, like interpretation as a double integral, require multiple steps to reduce the expression.

Consider, for instance, the antiderivative

∫ e − x 2 d x . \displaystyle\int e^{- x^2} \, dx. ∫e−x2dx.

This is known as the Gaussian integral, after its usage in the Gaussian distribution, and it is well known to have no closed form. However, the improper integral

I = ∫ 0 ∞ e − x 2 d x I = \int_0^\infty e^{- x^2} \, dx I=∫0∞​e−x2dx

may be evaluated precisely, using an integration trick. In fact, its value is given by the polar integral

I 2 = ∫ 0 ∞ ∫ 0 ∞ e − x 2 e − y 2 d y d x = ∫ 0 π / 2 ∫ 0 ∞ r e − r 2 d r d θ . I^2 = \int_0^\infty \int_0^\infty e^{-x^2} e^{-y^2} \, dy\, dx = \int_0^{\pi/2} \int_0^\infty r e^{-r^2} \, dr\, d\theta. I2=∫0∞​∫0∞​e−x2e−y2dydx=∫0π/2​∫0∞​re−r2drdθ.

Without such a method for exact evaluation of the integral, the Gaussian (normal) distribution would be significantly more complicated. Such integrals appear throughout physics, statistics, and mathematics.