How do you evaluate \inta^\infty f(x)\,dx?

\lim{t\to\infty}\inta^t f(x)\,dx.

What does it mean for an improper integral to converge?

The defining limit exists as a finite number.

Does \int1^\infty\frac1x\,dx converge?

No. \ln t\to\infty, so it diverges.

What if f is undefined inside [a,b]?

Split the integral at that point; it converges only if both parts converge.

Improper integrals and the mean value of a functionEdexcel A-Level Further Maths: Flashcards

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When is $\int_a^b f(x)\,dx$ improper?

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When is $\int_a^b f(x)\,dx$ improper?: If a limit is infinite, or $f$ is undefined at a value in $[a,b]$ .
How do you evaluate $\int_a^\infty f(x)\,dx$ ?: $\lim_{t\to\infty}\int_a^t f(x)\,dx$ .
What does it mean for an improper integral to converge?: The defining limit exists as a finite number.
Value of $\int_0^\infty e^{-x}\,dx$ ?: $1$ .
Does $\int_1^\infty\frac1x\,dx$ converge?: No. $\ln t\to\infty$ , so it diverges.
Value of $\int_1^\infty\frac{1}{x^2}\,dx$ ?: $1$ .
For which $p$ does $\int_1^\infty x^{-p}\,dx$ converge?: $p>1$ .
Value of $\int_0^2\frac{1}{\sqrt x}\,dx$ ?: $2\sqrt2$ .
What if $f$ is undefined inside $[a,b]$ ?: Split the integral at that point; it converges only if both parts converge.
Limit of $t\ln t$ as $t\to0^+$ ?: $0$ .
Limit of $te^{-t}$ as $t\to\infty$ ?: $0$ .
Mean value of $f$ over $[a,b]$ ?: $\frac{1}{b-a}\int_a^b f(x)\,dx$ .
Mean value of $x^2$ over $[1,4]$ ?: $7$ .

Exam questions on Improper integrals and the mean value of a function

Let $I=\int_0^\infty e^{-2x}\,dx$ and $J=\int_1^\infty \frac{1}{x}\,dx$ .
Use a limit to explain why $J$ has no finite value.2 marks
Let $f(x)=\frac{1}{\sqrt{x}}$ for $0<x\le 4$ . The function $f$ is not defined at $x=0$ .
Hence find the mean value of $f$ over the interval $0\le x\le4$ .2 marks
The temperature, $T\,^\circ$ C, in a greenhouse $t$ hours after 06:00 is modelled by $T=15+4\sin\left(\frac{\pi t}{12}\right)$ for $0\le t\le 12$ .
Find the exact mean temperature over the 12 hours.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).