FP3: IntegrationEdexcel International A Level Further Maths: Topic test

1. 1
   The identities $\cosh2x=1+2\sinh^2x$ and $\cosh2x=2\cosh^2x-1$ are used to integrate squares of hyperbolic functions.

   (a)

   Find $\int\sinh^2x\,dx$.

   [1 mark]

   • A$\frac14\sinh2x+\frac12x+c$
   • B$\frac13\sinh^3x+c$
   • C$\frac14\sinh2x-\frac12x+c$
   • D$\frac12\sinh2x-\frac12x+c$

   (b)

   Find the exact value of $\int_0^{\ln3}\sinh^2x\,dx$.

   [1 mark]

   • A$\frac{10}{9}-\frac12\ln3$
   • B$\frac{10}{9}+\frac12\ln3$
   • C$\frac{20}{9}-\frac12\ln3$
   • D$\frac{40}{9}-\frac12\ln3$

   (c)

   Find $\int\cosh^2x\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $K=\int_0^{\frac12}\arcsin x\,dx$.

   (a)

   Integration by parts is used with $u=\arcsin x$ and $\frac{dv}{dx}=1$, giving $K=\left[x\arcsin x\right]_0^{\frac12}-\int_0^{\frac12}f(x)\,dx$. Find $f(x)$.

   [1 mark]

   • A$\frac{1}{\sqrt{1-x^2}}$
   • B$x\sqrt{1-x^2}$
   • C$\frac{x}{1-x^2}$
   • D$\frac{x}{\sqrt{1-x^2}}$

   (b)

   Find the exact value of $\int_0^{\frac12}\frac{x}{\sqrt{1-x^2}}\,dx$.

   [1 mark]

   • A$\frac{\sqrt3}{2}-1$
   • B$1-\frac{\sqrt3}{2}$
   • C$\frac{\sqrt3}{2}$
   • D$\frac{2-\sqrt3}{4}$

   (c)

   Using the results above, find the exact value of $K$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^4\frac{dx}{\sqrt{x^2+9}}$.

   (a)

   Use the substitution $x=3\sinh u$ to show that $I=\ln3$.

   [3 marks]

   (b)

   Find the exact value of $\int_2^5\frac{dx}{\sqrt{x^2-4x+13}}$, giving your answer as a single logarithm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For integers $n\ge0$, let $I_n=\int_0^{\frac{\pi}{4}}\tan^nx\,dx$.

   (a)

   Show that $I_n+I_{n-2}=\frac{1}{n-1}$ for $n\ge2$. Hence find the exact value of $I_4$.

   [6 marks]

   (b)

   The region $R$ is bounded by the curve $y=\tan^3x$, the $x$-axis and the line $x=\frac{\pi}{4}$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis. Use the result of part (a) to find the exact volume of the solid formed, and give its value to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   For integers $n\ge0$, let $I_n=\int_0^1x^ne^x\,dx$.

   (a)

   Which of the following is correct for $n\ge1$?

   [1 mark]

   • A$I_n=e+nI_{n-1}$
   • B$I_n=e-nI_{n-1}$
   • C$I_n=nI_{n-1}-e$
   • D$I_n=e-I_{n-1}$

   (b)

   Find the value of $I_1$.

   [1 mark]

   • A$e-1$
   • B$e$
   • C$2e-1$
   • D$1$

   (c)

   Find the exact value of $I_2$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\cosh x$ for $0\le x\le\ln3$.

   (a)

   Find $1+\left(\frac{dy}{dx}\right)^2$.

   [1 mark]

   • A$\cosh^2x$
   • B$\sinh^2x$
   • C$1+\cosh^2x$
   • D$\cosh2x$

   (b)

   Find the length of $C$.

   [1 mark]

   • A$\frac53$
   • B$\frac23$
   • C$\frac43$
   • D$\frac{16}{9}$

   (c)

   The curve $C$ is rotated through $2\pi$ radians about the $x$-axis. Write down, but do not evaluate, an integral for the area of the surface generated, in terms of $x$ only.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The curve $C$ has parametric equations $x=t^2$, $y=2t$, for $0\le t\le2$.

   (a)

   Show that the length of $C$ is $\int_0^2 2\sqrt{1+t^2}\,dt$.

   [3 marks]

   (b)

   The curve $C$ is rotated through $2\pi$ radians about the $x$-axis. Find the exact area of the surface generated.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=\frac12x^2$ for $0\le x\le\frac34$.

   (a)

   Use the substitution $x=\sinh u$ to show that the length of $C$ is $\frac12\ln2+\frac{15}{32}$.

   [6 marks]

   (b)

   The curve $C$ is rotated through $2\pi$ radians about the $y$-axis. Find the exact area of the surface generated.

   [6 marks]

   Total for question 8: 12 marks

End of questions