Hyperbolic and trigonometric substitutionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A student evaluates integrals of the form $\int\frac{dx}{a^2+x^2}$ and $\int\frac{dx}{\sqrt{x^2\pm a^2}}$ using the standard results.

   (a)

   Find the exact value of $\int_0^3\frac{dx}{9+x^2}$.

   [1 mark]

   • A$\frac\pi4$
   • B$\frac{3\pi}{4}$
   • C$\frac\pi{12}$
   • D$\frac{\pi}{36}$

   (b)

   Find $\int\frac{dx}{\sqrt{x^2-4}}$ for $x>2$.

   [1 mark]

   • A$\operatorname{arsinh}\frac x2+c$
   • B$\operatorname{arcosh}\frac x2+c$
   • C$\arcsin\frac x2+c$
   • D$\frac12\operatorname{arcosh}\frac x2+c$

   (c)

   Find the exact value of $\int_0^3\frac{dx}{\sqrt{x^2+9}}$ in terms of a natural logarithm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I=\int\frac{dx}{\sqrt{x^2+6x+13}}$.

   (a)

   Which expression is equal to $x^2+6x+13$?

   [1 mark]

   • A$(x+3)^2+13$
   • B$(x+3)^2-4$
   • C$(x+6)^2-23$
   • D$(x+3)^2+4$

   (b)

   Find $I$.

   [1 mark]

   • A$\operatorname{arsinh}\frac{x+3}{2}+c$
   • B$\frac12\operatorname{arsinh}(x+3)+c$
   • C$\arcsin\frac{x+3}{2}+c$
   • D$\frac12\arctan\frac{x+3}{2}+c$

   (c)

   Hence find the exact value of $\int_{-3}^{1}\frac{dx}{\sqrt{x^2+6x+13}}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^2\frac{dx}{\left(x^2+4\right)^2}$.

   (a)

   Use the substitution $x=2\tan\theta$ to show that $I=\frac18\int_0^{\pi/4}\cos^2\theta\,d\theta$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Some integrals involving quadratic surds can be found using a hyperbolic or a trigonometric substitution.

   (a)

   Use the substitution $x=2\sinh u$ to find $\int\frac{x^2}{\sqrt{x^2+4}}\,dx$.

   [6 marks]

   (b)

   Use the substitution $x=\sin\theta$ to find the exact value of $\int_0^{1/2}\frac{x^2}{\sqrt{1-x^2}}\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions