Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

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Partial integration CDF

I am reading a textbook which claims that we can obtain by partial integration, for CDF $F(x)$:$$\int_{t}^{\infty} 1-F(x) \frac{dx}{x}=\int_{t}^{\infty} (\log u -\log t) dF(u) $$ I am aware that the latter integral is a Riemann-Stieltjes integral,…
Joogs
  • 417
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What is an integral other than "the area under the graph"?

What is a possible visual meaning of the integral of a real function $x(t)$ other than "the area under the graph"? I'm asking this so that i can avoid thinking about a graph when thinking about an integral, and view the integral as a property of a…
exp8j
  • 587
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Integrals of the form $\int \frac{f(x)}{x^p \pm 1}$ or $\int \frac{f(x)}{e^x\pm 1}$

I have been recently running into integrals of the form $$\int\frac{f(x)}{x^p \pm 1}dx$$ and $$\int \frac{f(x)}{e^x \pm 1} dx$$ where $f(x)$ is ususally $x^q$ or $\sin x$. For example, $f(x) = x^3$ occurs in context of Stefan's Law and Wien's Law. I…
dexter04
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$\int \frac{x^{2} \arctan x}{1+x^{2}}dx$

So I have the integral $$\int \frac{x^{2} \arctan x}{1+x^{2}}dx$$ how can I solve this integral without substituting $u=\arctan x$? Because I think that if I do this, lets suppose that in the end I'll have as a solution $\tan u + \cos u$ when I…
user60446
  • 79
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Alternative way to $\int \frac{dx}{1+x^3}$ .

$$\int \frac{dx}{1+x^3}$$ I know it is to be solved by breaking the denominator as $(1+x)(1-x+x^2)$ and then do partialization of the fraction. But I wanted to know if there is any different method to solve it . N. B. If the question is…
Entrepreneur
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proof that a function is integrable on a interval $[a,b]$

a) Divide a interval $[a,b]$ into $n$ equal subintervals. here I'm thinking $P_{n} =(x_0,x_1,x_2,x_3,x_{n-1}, x_n)$ where $a = x_0 < x_1 < x_2 < x_3 <\dots< x_{n-1} < x_n = b$ b) make an expression for the lower Riemann sum $L(f,P_{n})$ and the…
Jonas
  • 71
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What is the meaning of $g(dx)$ or $dg(x)$ in an integral

How can I interpret/solve an integration like: $$ \int f(x) \mathbf{d}g(x) $$ or $$ \int f(x) g(\mathbf{d}x) $$
Helium
  • 712
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Quick question why setting $a=0$ gives an indeterminate solution

$\newcommand{\dx}{\mathrm dx\,}$I’m having lots of trouble figuring this out, so perhaps you guys can help me. For example, let’s take the integral $$\int\limits_0^{\infty}\dx\frac {\sin x\log x}x=-\frac {\gamma\pi}2$$ Our integral can be computed…
Crescendo
  • 4,089
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Integral question rational functions?

I have to find the integral of $\frac{1}{(x+1)\sqrt{x^2+2x}}$. So I thought about writing it as $\frac{1}{(x+1)\sqrt{(x+1)^2-1}}$ here I replace $x+1=u$ and I have $\frac{1}{u(u^2-1)}$ . What do I do next? The answer on my textbook is…
erre
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Show that $\int_0^{2\pi}\int_0^t \frac r {\sqrt{t^2-r^2}} e^{-i k r \cos\theta} \, d\theta \, dr=2\pi \frac{\sin(k t)} k$

I'm trying to compute this integral:$$\int_0^{2\pi} \int_0^t \frac r {\sqrt{t^2-r^2}} e^{-i k r \cos\theta} \, d\theta \, dr$$ I can perform an integration by $\theta$ in terms of Bessel function, but maybe there is a simpler way?
Alexander Nikolaenko
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Problem with an integral

I'm trying to integrate this: $ U_{1}(Z_{1} + Z_{2},t) = C \int\nolimits_{0}^{\infty} (1-i\Phi) \exp[-(1+iV)g] \mathrm{d}g, $ with: $$ \Phi = \dfrac{\theta}{t_{c}} \int_{0}^{t} \dfrac{1}{1+2t'/t_{c}} \left( 1 - \exp \left[- \dfrac{2mg}{1+2t'/t_{c}}…
Rodrigo Thomas
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Are there any integrals that cannot be solved by a machine, but can be by hand?

My initial thoughts are no, since all you have to do is code all known integral techniques, and have the computer brute force the result. But would there be any integrals, or any integral techniques which would take less time to do by hand than by a…
AdronVenelucio
  • 173
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Integrate $\frac{\mathrm d x}{\sqrt{2x+1}}$ from 0 to 4

$$ \int_0^4 \frac{\mathrm d x}{\sqrt{2x+1}}$$ Then: $t=2x+1$ $\mathrm dt=2 \mathrm dx$ $\mathrm dx=\mathrm dt /2$ $$ \int_0^4 \frac{\mathrm d t}{2\sqrt t} $$ And what next?
Fre4kone
  • 113
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Integration problem: $\int_{0}^{1}\ln \left( \frac{e^{-\theta t}- 1}{e^{-\theta}-1} \right) \left(1 - e^{\theta t}\right)dt$

I'm trying to evaluate the following integral: $$\int_{0}^{1}\ln \left( \frac{e^{-\theta t}- 1}{e^{-\theta}-1} \right) \left(1 - e^{\theta t}\right)dt, ~~\theta \neq 0$$ However, I'm having trouble with it, not really sure how to tackle this. Any…
runr
  • 740
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How to compute $\int_0^\infty \frac{e^{-t}-1}{t^{s+1}}dt$

I came across the following integral in a textbook without explanation. How can I prove it? $$\int_0^\infty \frac{e^{-t}-1}{t^{s+1}}dt=\Gamma(-s)$$ Here $s\in(0,1)$.
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