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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double integral evaluation

Evaluate the integral $\int\int_R(x^2-2y^2) \, dA$ where $R$ is the first quadrant region between the circles of radius $4$ and radius $5$. I am having troubles with finding the limits of the integral. Thank you!
user131040
  • 597
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How can I find $\int_0^{\pi/2}x\cot x\,dx$?

Can $\displaystyle\int_0^{\pi/2}x\cot x\,dx$ be found using elementary functions? If so how could I possibly do it? Is there any other way to calculate above definite integral?
Mike
  • 31
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How find this integral $I=\int_{-\infty}^{+\infty}\frac{x^3\sin{x}}{x^4+x^2+1}dx$

Find this integral $$I=\int_{-\infty}^{+\infty}\dfrac{x^3\sin{x}}{x^4+x^2+1}dx$$ my idea: $$I=2\int_{0}^{+\infty}\dfrac{x^3\sin{x}}{x^4+x^2+1}dx$$ because…
math110
  • 93,304
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Find the volume generated by revolving the shaded region bounded by the given lines and curves about the y-axis.

The region enclosed by $x=\frac{y^2}{4}$, $x=0$, $y=-4$, and $y=4$. I know my limits are $0$, $4$. And I have the integral set up. But I'm having issues finding the anti-derivative of the functions.
Pia
  • 95
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Difficult integral: $\int e^x \ln (x)\, \mathrm dx$

I am trying to solve an integral: $\int e^x \ln(x)\,\mathrm dx =\ ?$ I have tried integration by parts and I found out that this method doesn't provide a solution in this case? How to solve it then?
Maartje
  • 31
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How to calculate $\int \frac{x}{\sqrt x -2}dx$

I don't know how to solve the following integral. I need some suggestions. Thank you! $$ \int \frac{x}{\sqrt x -2}dx$$
L_McClain
  • 523
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integrating a simple function

If $f(x) = \sum_{j=1}^n c_j 1_{A_j}(x)$ ie a simple function. Why is $$\prod_{j=1}^n \exp(t[\exp(i(u,c_j))-1]\mu(A_j)) =\exp[t\int_A[\exp(i(u,f(x)))-1]\mu(dx)].$$ The notation $(u,c)$ is dot product, $1_{A}(x)$ is the indicator function and…
Vaolter
  • 1,711
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Compute an indefinite integral of logarithms.

Is there a simple way of proving the following identity: \begin{eqnarray} \int \log(x) \log(x^2 + (x + W)^2) dx = \\ 2 x + \left(1 - \log(x)\right) \left(\log(e^{-\pi/2} W) W + 2 x + W \arctan(\frac{x+W}{x}) - (\frac{W}{2} + x)\log(x^2 + (x+…
Przemo
  • 11,331
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Knowing that $f(x)f(-x)=1$, evaluate the following integral

Knowing that $f(x)f(-x)=1$ for all $x$, evaluate the following integral: $$\int_{-\pi\over4}^{\pi\over4} {1\over{(1+2\sin^2x)(1+f(x))}} $$ Also, I found similar integrals of the form $\int_{-a}^{a} {1\over{(1+g(x))(1+f(x))}}$ where $g$ is even and…
Lisa
  • 733
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Integration Techniques - Adding [arbitrary] values to the numerator.

Suppose you wanted to evaluate the following integral. Where did the 4 come from? I understand that it makes the solution but how would you make an educated guess to put a 4? And how in the future would I solve similar questions?
Sad CRUD Developer
  • 452
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How find this $I=\int_{0}^{\infty}\frac{x\sin{(2x)}}{x^2+4}dx$

Find this integral $$I=\int_{0}^{\infty}\dfrac{x\sin{(2x)}}{x^2+4}dx$$ let $x=2t$, then $$I=\int_{0}^{\infty}\dfrac{t\sin{(4t)}}{(t^2+1)}dt$$ then $$I=1/2\int_{0}^{\infty}\sin{(4t)}d\ln{(t^2+1)}$$ then I can't. This problem have without residue …
math110
  • 93,304
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Integral of $n^x$

A simple high school math problem (It's been too long since I went there). I need to find the average of the yearly output when the degredation of the yearly output is given by $100 \times 0.98^x$. I need to be able to find the average between $x_1$…
Rune FS
  • 133
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Integral Of $\sqrt{9-y^2}dy$

I would like to integrate the following: $$\int\sqrt{9-y^2}dy$$ What I did: $y=3\sin t$ $dy=3\cos tdt$ $$\int\sqrt{9-9\sin ^2 t}\cdot 3\cos t dt=\int\sqrt{9}\cdot\sqrt{1-\sin^2 t}\cdot 3\cos tdt=\int 9\cos ^2tdt$$ $$9\cdot…
Ofir Attia
  • 3,136
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4 answers

Riemann-Integral in terms of trapeziums

Why isn't the Riemann integral evaluate in terms of trapeziums ? It would be a better method compared to rectangles.
Isomorphic
  • 1,182
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5 answers

Help with Evaluating triple integral

Evaluate $\int \int \int_B xyz^2 dV$, where B is a cuboid bounded by the regions $ 0 \le x \le 1 $, $ -1 \le y \le 2 $, $ 0 \le z \le 3 $. I keep getting $ \frac{27}{4}$ as my answer but apparently it's incorrect...Any help would be appreciated.
genius12
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