Improper integralsAQA A-Level Further Maths: Subtopic test

1. 1
   Consider the integrals $I=\int_0^4\dfrac{1}{\sqrt{x}}\,dx$ and $K=\int_0^4\dfrac{1}{x}\,dx$.

   (a)

   Which statement about $I$ and $K$ is correct?

   [1 mark]

   • ABoth $I$ and $K$ converge
   • B$I$ converges and $K$ diverges
   • C$K$ converges and $I$ diverges
   • DBoth $I$ and $K$ diverge

   (b)

   Find the value of $I$.

   [1 mark]

   • A$2$
   • B$\dfrac{16}{3}$
   • C$I$ diverges
   • D$4$

   (c)

   Show that $K$ diverges.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x\mathrm{e}^{-x}$ for $x\ge0$.

   (a)

   Find $\displaystyle\lim_{x\to\infty}\mathrm{f}(x)$.

   [1 mark]

   • A$\infty$
   • B$\dfrac{1}{\mathrm{e}}$
   • C$0$
   • D$1$

   (b)

   Find the coordinates of the stationary point of the curve $y=\mathrm{f}(x)$.

   [1 mark]

   • A$\left(1,\dfrac{1}{\mathrm{e}}\right)$
   • B$(1,\mathrm{e})$
   • C$(0,0)$
   • D$\left(-1,-\mathrm{e}\right)$

   (c)

   Find the exact value of $\int_0^\infty\mathrm{f}(x)\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^1\ln x\,dx$.

   (a)

   Show that, for $0<t<1$, $\int_t^1\ln x\,dx=-1+t-t\ln t$.

   [3 marks]

   (b)

   Hence evaluate $I$, and explain why $I$ is an improper integral.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For a real constant $p$, consider the integral $I(p)=\int_1^\infty x^{-p}\,dx$.

   (a)

   (i) Show that, for $p>1$, $I(p)=\dfrac{1}{p-1}$.
   (ii) Show that $I(1)$ does not converge.

   [6 marks]

   (b)

   (i) Find the value of $p$ for which $I(p)=4$.
   (ii) For $p<1$, $\int_0^1x^{-p}\,dx$ converges and equals $\dfrac{1}{1-p}$. Hence explain why $\int_0^\infty x^{-p}\,dx$ diverges for every real value of $p$.

   [6 marks]

   Total for question 4: 12 marks

End of questions