How do you check an indefinite integral?

Differentiate your answer; you should get the original function.

Integrating standard functionsEdexcel A-Level Maths: Flashcards

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$\int e^{kx}\,\mathrm{d}x$?

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$\int e^{kx}\,\mathrm{d}x$ ?: $\frac1ke^{kx}+c$
$\int\frac1x\,\mathrm{d}x$ ?: $\ln|x|+c$
$\int\frac{1}{2x}\,\mathrm{d}x$ ?: $\frac12\ln|x|+c$
$\int\sin kx\,\mathrm{d}x$ ?: $-\frac1k\cos kx+c$
$\int\cos kx\,\mathrm{d}x$ ?: $\frac1k\sin kx+c$
$\int\sec^2kx\,\mathrm{d}x$ ?: $\frac1k\tan kx+c$
Identity used to integrate $\sin^2x$ ?: $\sin^2x=\frac12(1-\cos2x)$
Identity used to integrate $\cos^2x$ ?: $\cos^2x=\frac12(1+\cos2x)$
Identity used to integrate $\tan^2x$ ?: $\tan^2x=\sec^2x-1$
$\int\tan^2x\,\mathrm{d}x$ ?: $\tan x-x+c$
$\int e^{5x}\,\mathrm{d}x$ ?: $\frac15e^{5x}+c$
What units must the angle be in when you integrate trigonometric functions?: Radians.
How do you check an indefinite integral?: Differentiate your answer; you should get the original function.

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).