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.

73636 questions
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Integration involving beta and hypergeometric functions/ differential binomial

So, hey, everybody! I have to integrate this $$ \int_0^2 \sqrt[3]{\frac{x^2}{2-x}} \, dx $$ and I've already figured out that due to Chebyshev's theorem it cannot be done in terms of elementary functions, since we can rewrite the task as $$…
Lebesgue
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Can One Integrate $\frac{1}{i} \int \frac{e^{ix}-e^{-ix}}{e^{ax}+e^{-ax}+e^{ix}+e^{-ix}}dx$

I'd like to integrate $$I(a)=\int \frac{\sin(x)}{\cosh(ax)+\cos(x)}dx$$ by changing $sin$ etc... into their exponential representation. Using $e^{ix} = \cos(x) + i \sin(x)$ and $e^{ax} = \cosh(ax)+\sinh(ax)$ we have $$\frac{1}{i} \int…
bolbteppa
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$\tan(x)$ substitution in these kind of integrals

I heard that we can use $\tan(x)=y$ substitution in integrals where both sine and cosine is on even power like: $\int \sin^2(x)\cdot \cos^2(x)\, dx$. How exactly can I use it? I know that we can solve it in other way, but I want to see how this…
user128576
  • 357
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Judge the convergence of general integral

$$\int_{0}^{1} \frac{1}{e^{\sqrt{x}}-1}dx$$ Prove it is convergent or divergent. The main problem I face is how to deal with the expotential function in such a position.
zhen
  • 101
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Area of the circle.

I want to calculate the area of the circle of radius $\mathfrak{R}$. I would like to do it using the Cartesian coordinates (not the polar ones). The problem is that I found the area of a circle of center (0, 0) and radius $\mathfrak{R}$ is…
user153714
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Integration - area between $y=5a^2+4ax-x^2$ and $y=x^2-a^2$

So part a I've done, and it's $(3a, 8a^2)$ and $(-a,0)$ I've tried integrating it to find the area: $\int_{-a}^{3a} 3a^2+2ax-x^2 \ dx =…
Jim
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Area of graph $y=x^n$

I suppose you could solve this algebraically, but apparently, there is a way to solve this without doing anything to the red area to do this question. Can someone explain how?
Jim
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How to find anti-derivative of $f(x):=\sin(x)\cos(nx)$.

How to find anti-derivative of $f(x):=\sin(x)\cos(nx)$. I know the sum-formulas for $\sin(x),\cos(x)$ which I could put to use if $x = nx$, but this is not the case here where $n \in \mathbb N$. Any suggestions ?
Shuzheng
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$\int\tan x\ dx$ by integration by parts

Can anyone help me figure out what is wrong in the following step: $$\begin{align}\int \tan x\ dx &= \int (\sec x \sin x)\ dx\\ &= -\sec x \cos x + \int \sec x\tan x\cos x\ dx\\ &= -1 + \int \tan x dx\end{align}$$ So I got $\displaystyle\int \tan x\…
Jenny
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Show that $f\colon\bar{M}\to\mathbb{R}$ is Riemann-integrable

Consider a Jordan-measurable set $M\subset\mathbb{R}^n$ and $f\colon\bar{M}\to\mathbb{R}$ continious. Show that $f$ is Riemann-integrable over $M$. I think the main facts for the prove are: $M$ is bounded and so $\bar{M}$ is compact $f$…
user34632
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Integration by subtitution

Can someone explain me how to find the value of $$L = \int_1^2 \sqrt{1+9x} \,\mathrm{d}x$$ I do not know how to approach it after having $z = 1+9x$ and $\mathrm{d}x = \mathrm{d}z/9$.
Peter
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Is the following integral equation true?

I am reading a script and I found the following statement: $$ \int_{-\infty}^\infty \frac{1}{\sigma \sqrt{2\pi}} \exp \left(\frac{-x^2}{2 \sigma^2 }\right) \exp(i \, x\, \xi) \,dx = \exp\left(-\frac{1}{2} \xi^2 \sigma^2\right) $$ I tried to check…
Adam
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Solution for divergent integral

The integral $I = \int\limits_0^\infty \mathrm{d}x \, x \sin(x)$ does not converge. In physics we often use the principle of a convergence generating factor, in this example $I = \lim\limits_{\epsilon \to 0} \int\limits_0^\infty \mathrm{d}x \, x…
DaPhil
  • 208
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Area calculation of ellipse $x^2/2+y^2=1$

Calculate the area of the ellipse that you get when you rotate the ellipse $$\frac{x^2}{2}+y^2= 1$$ around the x-axis. My approach has been to use the formula for rotation area from $-2$ to $2$. But this gives a complicated integral and I'm unsure…
iveqy
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Integration with Euler number and sin2x

I found this example in a textbook: $$\int e^{\cos^2 x}\sin2x dx$$ There are also results, but I am not even close to that...
user984621
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