When does an improper integral converge?

When the limit defining it exists and is finite.

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

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

What if the undefined point is inside the range?

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

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

No: \ln t\to\infty as t\to\infty.

Improper integralsAQA A-Level Further Maths: Flashcards

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Two ways an integral can be improper?

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Two ways an integral can be improper?: An infinite limit of integration, or an integrand undefined at a point in the range.
When does an improper integral converge?: When the limit defining it exists and is finite.
How do you evaluate $\int_a^\infty\mathrm{f}(x)\,dx$ ?: $\lim_{t\to\infty}\int_a^t\mathrm{f}(x)\,dx$
How do you evaluate $\int_0^b\mathrm{f}(x)\,dx$ if $\mathrm{f}$ is undefined at $0$ ?: $\lim_{t\to0^+}\int_t^b\mathrm{f}(x)\,dx$
What if the undefined point is inside the range?: Split the integral at that point; it converges only if both parts converge.
$\int_1^\infty x^{-p}\,dx$ for which $p$ does it converge, and to what?: $p>1$ , to $\frac{1}{p-1}$ .
Value of $\int_1^\infty\frac{1}{x^2}\,dx$ ?: $1$
Does $\int_1^\infty\frac1x\,dx$ converge?: No: $\ln t\to\infty$ as $t\to\infty$ .
Limit of $x^k\mathrm{e}^{-x}$ as $x\to\infty$ , $k>0$ ?: $0$
Limit of $x^k\ln x$ as $x\to0^+$ , $k>0$ ?: $0$
Value of $\int_0^1\ln x\,dx$ ?: $-1$
Value of $\int_0^\infty x\mathrm{e}^{-x}\,dx$ ?: $1$

Exam questions on Improper integrals

Consider the integrals $I=\int_0^4\dfrac{1}{\sqrt{x}}\,dx$ and $K=\int_0^4\dfrac{1}{x}\,dx$ .
Show that $K$ diverges.2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x\mathrm{e}^{-x}$ for $x\ge0$ .
Find the exact value of $\int_0^\infty\mathrm{f}(x)\,dx$ .2 marks
Let $I=\int_0^1\ln x\,dx$ .
Show that, for $0<t<1$ , $\int_t^1\ln x\,dx=-1+t-t\ln t$ .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).