Improper integralsAQA A-Level Further Maths: Mind map

Exam questions on Improper integrals

1. Consider the integrals $I=\int_0^4\dfrac{1}{\sqrt{x}}\,dx$ and $K=\int_0^4\dfrac{1}{x}\,dx$.
   Show that $K$ diverges.2 marks
2. The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x\mathrm{e}^{-x}$ for $x\ge0$.
   Find the exact value of $\int_0^\infty\mathrm{f}(x)\,dx$.2 marks
3. Let $I=\int_0^1\ln x\,dx$.
   Show that, for $0<t<1$, $\int_t^1\ln x\,dx=-1+t-t\ln t$.3 marks