Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 4

The fundamental theorem of calculus differentiates the upper endpoint, while differentiation under the integral sign differentiates the parameter:These formulas hold, for example, when $g$ and its displayed parameter partial derivative are continuous on a neighborhood of the integration interval and the parameter value.

Work first with $|k|<1$, so the real integrands and their parameter derivatives are smooth on every finite amplitude interval. For the elliptic integral of the second kind,On the other hand, subtracting the integrands of the second-kind and first-kind elliptic integrals givesConsequentlyThis is the modulus derivative of the incomplete elliptic integral of the second kind. At $k=0$ its right-hand side has a removable limit zero, agreeing with $E_k(\theta ,0)=0$. Both $E$ and $F$ here use amplitude $\theta$ and modulus $k$; differentiating with respect to the alternative parameter $k^2$ would give a different prefactor.

The fundamental theorem of calculus and part (a) giveThe difference is constant in $\theta$. Since $I(0,k)=E(0,k)=0$, the constant is zero, provingThis is simply the chain rule integrated, with $E$ itself as the antiderivative of the square-root factor.