Why do we write \ln|x| and not \ln x for \int\frac1x\,dx?

The integral is valid for negative x too, and \ln needs a positive argument.

What does \inta^b\frac{dy}{dx}\,dx give?

The change in y from x=a to x=b.

Integrating standard functionsEdexcel International A Level Maths: Flashcards

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$\int e^{kx}\,dx$

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$\int e^{kx}\,dx$: $\frac1ke^{kx}+c$
$\int\frac1x\,dx$: $\ln|x|+c$ , for $x\neq0$
$\int\cos kx\,dx$: $\frac1k\sin kx+c$
$\int\sin kx\,dx$: $-\frac1k\cos kx+c$
$\int a^x\,dx$: $\frac{a^x}{\ln a}+c$
$\int6e^{2x}\,dx$: $3e^{2x}+c$
$\int\frac{1}{2x}\,dx$: $\frac12\ln|x|+c$
$\int\sin3x\,dx$: $-\frac13\cos3x+c$
$\int2\cos\frac x2\,dx$: $4\sin\frac x2+c$
$\int3e^{-0.5t}\,dt$: $-6e^{-0.5t}+c$
$\int2^x\,dx$: $\frac{2^x}{\ln2}+c$
Why do we write $\ln|x|$ and not $\ln x$ for $\int\frac1x\,dx$ ?: The integral is valid for negative $x$ too, and $\ln$ needs a positive argument.
What does $\int_a^b\frac{dy}{dx}\,dx$ give?: The change in $y$ from $x=a$ to $x=b$ .

Exam questions on Integrating standard functions

The function $f$ is defined by $f(x)=6e^{2x}-\frac{3}{x}$ for $x>0$ .
The curve $y=F(x)$ has gradient $f(x)$ and passes through the point $(1,3e^2+5)$ . Find $F(x)$ .2 marks
The function $g$ is defined by $g(x)=4\sin 3x+2\cos\frac{x}{2}$ , where $x$ is in radians.
Given that $\frac{dF}{dx}=g(x)$ and $F(0)=0$ , find the exact value of $F(\pi)$ .2 marks
A particle moves along a straight line. Its velocity is $v=3e^{-0.5t}+2$ m s $^{-1}$ at time $t$ seconds, for $t\geq0$ , and its position relative to a fixed point $O$ is $s$ metres.
Find the displacement of the particle between $t=0$ and $t=4$ , giving your answer to 3 significant figures.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).