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 with square root involved

How to compute the integral of $\sqrt{1-r^2}$ with respect to $r$? Is there a substitution? What are the steps? Thanks
Derk
  • 277
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Prove the equality $\frac{2}{\pi}\int_0^{\infty}\frac{\cos tx}{1+x^2}dx=e^{-|t|}$

Can somebody remind me how to prove the equality $$\frac{2}{\pi}\int_0^{\infty}\frac{\cos tx}{1+x^2}dx=e^{-|t|}?$$ I found this in a book, where it is considered as obvious. As always, it is obvious once you know the trick, so could somebody remind…
yannis
  • 235
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How to calculate this area in $\mathbb{R}^2$?

Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$ First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?). Here is what I did for the first…
x.y.z...
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Integration step at once without working out parentheses

In a solutions manual I see they integrate this $\frac{1}{2}r(4-r^2)^2$ and in the next this is $-\frac{1}{12}(4-r^2)^3$. Is this possible without working out the parentheses? Can someone explain this step? Thanks!
Derk
  • 277
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Ap calculus question (gr.12)

let $f(k)$ be a function determined as the average value of $h(x) = sin(x-k)$ on the interval $[0,\pi]$. Show that $f$ is a continuous function of $k$ and determine the maximum and minimum values of $f$.
breaane
  • 23
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Solution of a integral

$$ \int e^x \, \left(1 + \frac{e^{-x}}{x} \right) \,dx $$ I got three different integrals from this one, which are integral of $e^x$, integral of $1/x$ and the third one is integral of $e^{-x}/x$ but I'm not sure how to solve the third one? Thanks…
user133022
  • 49
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How to evaluate the integral

Can some one provide me hint to evaluate the following integral. $$ \int\csc^{2}\left(x\right) \ln\left(\cos\left(x\right) - \sqrt{\vphantom{\largeA}\,\cos\left(2x\right)\,}\,\right) \,{\rm d}x $$.
Kumar
  • 2,259
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How to evaluate $\int_{0}^{\infty}t^{n}e^{-at^{2}}dt$?

How to evaluate the integral $\int_{0}^{\infty}t^{n}e^{-at^{2}}dt$ where n is a positive even number and $\int_{0}^{\infty}e^{-at^{2}}dt=\frac{1}{2}\sqrt{\pi /a}$
esege
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integral $\int_k^\infty e^{ax}f(x) dx$ where $f$ is normal density

Calculate $$\int_k^\infty e^{ax}f(x) dx$$ where $f$ is the probability density of the normal distribution, i.e. $f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma^2}}$. I tried calculating the integral for a general integrable…
user126540
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Cartesian to polar coordinates - Integration

$$\iint_R \frac{1}{1+x^2+y^2} \,dA$$ $$R=\left\{(r,\theta):1\le r\le 2,0\le \theta \le \pi\right\}$$ limits of outer integral are $0$ to $\pi$ and inner integral are $1$ to $2$. I wanted to confirm if i did the problem right. My answeR:…
powerservice
  • 21
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Find $\int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx$ ($A$ is constant)

How does one evaluate the integral $$\int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx $$ where $A$ is a constant? Thanks to Perry Iverson，Steven Stadnicki，Jack D'Aurizio for my former question by Taylor series. However, multiplying by a constant A, can it…
Ison
  • 95
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Integrate $\frac{1}{\sqrt{6x-x^2-5}}$

Question: Integrate $\frac{1}{\sqrt{6x-x^2-5}}$ My Working: $$\int{\frac{1}{\sqrt{6x-x^2-5}}} = \int{\frac{1}{\sqrt{-(x^2-6x+5)}}} = \int{\frac{1}{\sqrt{-((x-3)^2-3^2+5)}}} = \int{\frac{1}{\sqrt{-((x-3)^2-4)}}}$$ Is it right so far? Because I think…
Jiew Meng
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Integrate $(\cos(x)-e^{-x^2})/x$

How does one integrate the following? $$\int_{0}^{\infty}\frac{\cos(x)-e^{-x^2}}{x} dx$$ Is it possible to do this without using mellintransformation?
Parseval
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Following definite integral

Here is the integral: $$\int_{0}^{2}\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} dx$$ Here is my work: $$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}} := y \implies x=y^2-y$$ By implicit differentiation, $$1 = 2y\frac{dy}{dx}-\frac{dy}{dx} \implies…
user85362
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Fallacy in integration technique to find a sphere's surface area?

The problem is to find the surface area of a sphere. Consider the first approach as shown in the following picture (the approach is used in Wikipedia too): In this first approach, the sphere is divided into infinitesimal rings where a ring has a…
Tadeus Prastowo
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