it can easily be shown that Given , this integral reduction formula can be used to compute all of the values of for . Integrals like and may also be handled using the Weierstrass substitution. Example 5 Here, we consider the integral Differentiating under the integral with respect to , we have Therefore: But by definition so and Example 6 Here, we consider the integral We introduce a new variable φ and rewrite the integral as When φ = 1 this equals the original integral. However, this more general integral may be differentiated with respect to : Now, fix φ, and consider the vector field on defined by . Further, choose the positive oriented parametrization of the unit circle given by , , so that . Then the final integral above is precisely the line integral of over . By Green's Theorem, this equals the double integral where is the closed unit disc. Its integrand is identically 0, so is likewise identically zero. This implies that f(φ) is constant. The constant may be determined by evaluating at : Therefore, the original integral also equals . Other problems to solve There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter : The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when . Therefore, differentiation under the integral sign is easy to justify when , but proving that the resulting formula remains valid when requires some careful work. Infinite series The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence. In popular culture Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman! in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using this technique, Feynman was able to