-\ln|\cos x|+c, or \ln|\sec x|+c

What should you do if the derivative of your guess is a multiple of the integrand?

Divide (or multiply) by that constant.

Integration by recognition and trigonometric identitiesEdexcel International A Level Maths: Flashcards

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$\int\frac{f'(x)}{f(x)}\,dx$

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$\int\frac{f'(x)}{f(x)}\,dx$: $\ln|f(x)|+c$
$\int f'(x)[f(x)]^n\,dx$ for $n\neq-1$: $\frac{[f(x)]^{n+1}}{n+1}+c$
$\int\tan x\,dx$: $-\ln|\cos x|+c$ , or $\ln|\sec x|+c$
$\int\sec^22x\,dx$: $\frac12\tan2x+c$
$\int\frac{x}{x^2+5}\,dx$: $\frac12\ln(x^2+5)+c$
$\int\frac{2}{2x-1}\,dx$: $\ln|2x-1|+c$
$\int\frac{2}{(2x-1)^4}\,dx$: $-\frac{1}{3(2x-1)^3}+c$
Identity for $\cos^2x$ ?: $\cos^2x=\frac12(1+\cos2x)$
Identity for $\sin^2x$ ?: $\sin^2x=\frac12(1-\cos2x)$
Identity used to integrate $\tan^2x$ ?: $\tan^2x=\sec^2x-1$
$\int\tan^2x\,dx$: $\tan x-x+c$
$\int\cos^23x\,dx$: $\frac x2+\frac{\sin6x}{12}+c$
What should you do if the derivative of your guess is a multiple of the integrand?: Divide (or multiply) by that constant.

Exam questions on Integration by recognition and trigonometric identities

The function $h$ is defined by $h(x)=\frac{x}{x^2+5}$ for $x\geq0$ .
Find $\int\frac{x}{(x^2+5)^2}\,dx$ .2 marks
The function $f$ is defined by $f(x)=\tan x$ for $0\leq x<\frac{\pi}{2}$ .
Find $\int[f(x)]^2\,dx$ .2 marks
The curves $C_1$ and $C_2$ have equations $y=\cos^23x$ and $y=\sin^23x$ respectively, for $0\leq x\leq\frac{\pi}{12}$ .
Show that $\cos^23x=\frac12(1+\cos6x)$ , and hence find $\int\cos^23x\,dx$ .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).