Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 20 / c / i / Solution

Write $I(x)=\ln g(x)$. For $x>-1$, differentiation under the integral sign givesIndeed, on any compact parameter interval $c\le x\le d$ with $c>-1$, the differentiated integrand is bounded by the integrable function $t^c$. The dominated convergence theorem therefore justifies the operation despite the improper lower endpoint. The apparent singularity at $t=1$ is removable.

Since $g=e^I$, the chain rule gives the first-order linear differential equation