\int\frac{dx}{\sqrt{a^2+x^2}}?

\operatorname{arsinh}\frac xa+c

\int\frac{dx}{\sqrt{x^2-a^2}}?

\operatorname{arcosh}\frac xa+c, for xa

What must you change after a substitution in a definite integral?

The limits, to values of the new variable.

First step for \int\frac{dx}{\sqrt{x^2+6x+13}}?

Complete the square: (x+3)^2+4.

How do you integrate \sin^2\theta?

Use \sin^2\theta=\frac12\left(1-\cos2\theta\right).

Hyperbolic and trigonometric substitutionsEdexcel International A Level Further Maths: Flashcards

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$\int\frac{dx}{a^2+x^2}$?

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$\int\frac{dx}{a^2+x^2}$ ?: $\frac1a\arctan\frac xa+c$
$\int\frac{dx}{\sqrt{a^2-x^2}}$ ?: $\arcsin\frac xa+c$
$\int\frac{dx}{\sqrt{a^2+x^2}}$ ?: $\operatorname{arsinh}\frac xa+c$
$\int\frac{dx}{\sqrt{x^2-a^2}}$ ?: $\operatorname{arcosh}\frac xa+c$ , for $x>a$
$\operatorname{arsinh}x$ in logarithmic form?: $\ln\left(x+\sqrt{x^2+1}\right)$
Substitution for $a^2+x^2$ ?: $x=a\tan\theta$
Substitution for $\sqrt{a^2-x^2}$ ?: $x=a\sin\theta$
Substitution for $\sqrt{a^2+x^2}$ ?: $x=a\sinh u$
Substitution for $\sqrt{x^2-a^2}$ ?: $x=a\cosh u$
What must you change after a substitution in a definite integral?: The limits, to values of the new variable.
First step for $\int\frac{dx}{\sqrt{x^2+6x+13}}$ ?: Complete the square: $(x+3)^2+4$ .
$\int\frac{dx}{x^2+4x+13}$ ?: $\frac13\arctan\frac{x+2}{3}+c$
How do you integrate $\sin^2\theta$ ?: Use $\sin^2\theta=\frac12\left(1-\cos2\theta\right)$ .

Exam questions on Hyperbolic and trigonometric substitutions

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).