Integration by recognition and trigonometric identitiesEdexcel International A Level Maths: Revision notes

If the numerator is the derivative of the denominator, the integral is a logarithm: $\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+c.$ Examples: $\int\frac{x}{x^2+5}dx=\frac12\int\frac{2x}{x^2+5}dx=\frac12\ln(x^2+5)+c$. Also $\int\frac{2}{2x-1}dx=\ln|2x-1|+c$. Tangent: $\tan x=\frac{\sin x}{\cos x}$ and $\frac{d}{dx}\cos x=-\sin x$, so $\int\tan x\,dx=-\ln|\cos x|+c=\ln|\sec x|+c.$

If the integrand is a derivative times a power of the function, use the power rule in reverse: $\int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+c,\qquad n\neq-1.$ Examples: $\int\frac{x}{(x^2+5)^2}dx=-\frac{1}{2(x^2+5)}+c$. And $\int\frac{2}{(2x-1)^4}dx=-\frac{1}{3(2x-1)^3}+c$. For a function of $2x$ such as $\sec^22x$: $\frac{d}{dx}\tan2x=2\sec^22x$, so $\int\sec^22x\,dx=\frac12\tan2x+c$.

If the derivative of your guess differs from the integrand by a constant factor, multiply or divide to match. Example: guess $\ln(x^2+5)$. Its derivative is $\frac{2x}{x^2+5}$, which is twice $\frac{x}{x^2+5}$, so the correct integral is $\frac12\ln(x^2+5)$. If $n=-1$ the power rule fails and the integral becomes a logarithm, as in the previous section.

Squares of trigonometric functions need an identity first:

• $\sin^2x=\frac12(1-\cos2x)$
• $\cos^2x=\frac12(1+\cos2x)$
• $\tan^2x=\sec^2x-1$ Examples: $\int\sin^2x\,dx=\frac x2-\frac{\sin2x}{4}+c$; $\int\cos^23x\,dx=\int\frac12(1+\cos6x)\,dx=\frac x2+\frac{\sin6x}{12}+c$; $\int\tan^2x\,dx=\tan x-x+c$.

Apply limits after integrating. Use exact values such as $\tan\frac\pi4=1$, $\cos\frac\pi3=\frac12$. Example: $\int_0^{\pi/4}\tan^2x\,dx=\left[\tan x-x\right]_0^{\pi/4}=1-\frac\pi4$. Example: $\int_0^{\pi/12}(\cos^23x-\sin^23x)\,dx=\int_0^{\pi/12}\cos6x\,dx=\frac16$, using $\cos2A=\cos^2A-\sin^2A$. Area between curves is $\int(\text{upper}-\text{lower})\,dx$. For a curve under a horizontal line, use a rectangle minus the area under the curve.