Integrating hyperbolic and inverse functionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A student is working with the integral $I=\int_0^{\ln2}\cosh2x\,dx$.

   (a)

   Find $\int\cosh2x\,dx$.

   [1 mark]

   • A$2\sinh2x+c$
   • B$\frac12\sinh2x+c$
   • C$\sinh2x+c$
   • D$\frac12\cosh2x+c$

   (b)

   Find the exact value of $I$.

   [1 mark]

   • A$\frac{15}{8}$
   • B$\frac{17}{16}$
   • C$\frac38$
   • D$\frac{15}{16}$

   (c)

   Hence find the exact value of $\int_0^{\ln2}\sinh^2x\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Integration by parts is used to integrate the product of $x$ with a hyperbolic function.

   (a)

   Find $\int x\cosh x\,dx$.

   [1 mark]

   • A$x\sinh x-\cosh x+c$
   • B$x\sinh x+\cosh x+c$
   • C$x\sinh x-\sinh x+c$
   • D$x\cosh x-\sinh x+c$

   (b)

   Find the exact value of $\int_0^1x\cosh x\,dx$.

   [1 mark]

   • A$e-1$
   • B$-e^{-1}$
   • C$1-e^{-1}$
   • D$e^{-1}-1$

   (c)

   Find $\int x\sinh x\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^{4/3}\operatorname{arsinh}x\,dx$.

   (a)

   Use integration by parts to find $\int\operatorname{arsinh}x\,dx$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $I=\int_0^1\arctan x\,dx$ and $J=\int_0^1x\arctan x\,dx$.

   (a)

   Show that $I=\frac\pi4-\frac12\ln2$.

   [6 marks]

   (b)

   Find the exact value of $J$.

   [6 marks]

   Total for question 4: 12 marks

End of questions