Integrating standard functionsEdexcel A-Level Maths: Revision notes

Section 1

Reversing differentiation

Integration is the reverse of differentiation, so every standard derivative gives a standard integral. Always add the constant $c$ to an indefinite integral. Integrals of sums, differences and constant multiples are found term by term: $\int\big(af(x)\pm bg(x)\big)\mathrm{d}x=a\int f(x)\,\mathrm{d}x\pm b\int g(x)\,\mathrm{d}x.$ The standard results you need are $\int e^{kx}\,\mathrm{d}x=\frac1ke^{kx}+c$ , $\int\frac1x\,\mathrm{d}x=\ln|x|+c$ , $\int\sin kx\,\mathrm{d}x=-\frac1k\cos kx+c$ , $\int\cos kx\,\mathrm{d}x=\frac1k\sin kx+c$ and $\int\sec^2kx\,\mathrm{d}x=\frac1k\tan kx+c$ .

Key termsindefinite integralconstant of integration

Exam tip

Check any integral by differentiating your answer: you must get back the original function.

Section 2

Exponentials and reciprocals

For $e^{kx}$ the factor $\frac1k$ undoes the chain-rule factor $k$ : $\int e^{5x}\,\mathrm{d}x=\frac15e^{5x}+c$ and $\int6e^{2x}\,\mathrm{d}x=3e^{2x}+c$ . For $\frac1x$ the result is a logarithm, not a power. Rewrite $\frac{1}{2x}$ as $\frac12\times\frac1x$ , so $\int\frac{1}{2x}\,\mathrm{d}x=\frac12\ln|x|+c$ . The same idea gives $\int\frac{4}{x}\,\mathrm{d}x=4\ln|x|+c$ . Worked example: $\frac{\mathrm{d}y}{\mathrm{d}x}=6e^{2x}-\frac4x$ through $(1,3e^2+1)$ gives $y=3e^{2x}-4\ln x+c$ , and $x=1$ gives $c=1$ .

Key termsnatural logarithm

Common mistake

Using the power rule on $x^{-1}$ : it would divide by zero. $\int x^{-1}\,\mathrm{d}x=\ln|x|+c$ .

Common mistake

Writing $\int\frac{1}{2x}\,\mathrm{d}x=\ln|2x|+c$ : differentiating that gives $\frac1x$ , not $\frac1{2x}$ .

Section 3

Trigonometric functions

With $x$ in radians: $\int\sin3x\,\mathrm{d}x=-\frac13\cos3x+c$ , $\int\cos\frac x2\,\mathrm{d}x=2\sin\frac x2+c$ and $\int\sec^22x\,\mathrm{d}x=\frac12\tan2x+c$ . The sign matters: $\frac{\mathrm{d}}{\mathrm{d}x}(\cos x)=-\sin x$ , so integrating $\sin kx$ introduces a minus sign, but integrating $\cos kx$ does not. Since $\frac{\mathrm{d}}{\mathrm{d}x}(\tan x)=\sec^2x$ , any $\sec^2kx$ integrates to $\frac1k\tan kx$ .

Key termsradians

Common mistake

Forgetting the minus sign when integrating $\sin kx$ .

Common mistake

Multiplying by $k$ instead of dividing by $k$ .

Section 4

Using identities before integrating

Powers of sine, cosine and tangent cannot be integrated directly, so rewrite them first: $\sin^2x=\tfrac12(1-\cos2x),\quad\cos^2x=\tfrac12(1+\cos2x),\quad\tan^2x=\sec^2x-1.$ Worked example: $\cos^23x=\frac12(1+\cos6x)$ , so $\int\cos^23x\,\mathrm{d}x=\frac x2+\frac{\sin6x}{12}+c$ . Similarly $\int\tan^2x\,\mathrm{d}x=\int(\sec^2x-1)\,\mathrm{d}x=\tan x-x+c$ .

Key termsdouble-angle identity

Exam tip

For $\cos^2kx$ the new angle is $2kx$ , so the integrated term is divided by $2k$ .

Section 5

Definite integrals with exact values

For a definite integral, substitute the limits into the integrated function and subtract: $\int_a^bf(x)\,\mathrm{d}x=F(b)-F(a)$ . Leave answers exact ( $\pi$ , $\ln2$ , surds) unless a decimal is asked for. Example: $\int_0^{\pi}\left(\sin3x+\cos\frac x2\right)\mathrm{d}x=\left[-\frac13\cos3x+2\sin\frac x2\right]_0^{\pi}=\frac23+2=\frac83$ .

Key termsdefinite integral

Common mistake

Setting the calculator to degrees: calculus with trigonometric functions needs radians.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Integrating standard functions

A curve has gradient $\frac{\mathrm{d}y}{\mathrm{d}x}=6e^{2x}-\frac{4}{x}$ for $x>0$ , and passes through the point $(1,\,3e^2+1)$ .
Find the equation of the curve.2 marks
Angles are in radians. A function is defined by $f(x)=\sin3x+\cos\frac{x}{2}$ .
Find the exact value of $\int_0^{\pi}f(x)\,\mathrm{d}x$ .2 marks
Angles are in radians. Trigonometric identities are used to rewrite an expression before it is integrated.
Show that $\int\sin^2x\,\mathrm{d}x=\frac{x}{2}-\frac{\sin2x}{4}+c$ .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).