Hyperbolic and trigonometric substitutionsEdexcel International A Level Further Maths: Mind map

Exam questions on Hyperbolic and trigonometric substitutions

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.
   Find the exact value of $\int_0^3\frac{dx}{\sqrt{x^2+9}}$ in terms of a natural logarithm.2 marks
2. Let $I=\int\frac{dx}{\sqrt{x^2+6x+13}}$.
   Hence find the exact value of $\int_{-3}^{1}\frac{dx}{\sqrt{x^2+6x+13}}$.2 marks
3. Let $I=\int_0^2\frac{dx}{\left(x^2+4\right)^2}$.
   Use the substitution $x=2\tan\theta$ to show that $I=\frac18\int_0^{\pi/4}\cos^2\theta\,d\theta$.3 marks