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
3
votes
2 answers

how can i change specifically the intervals of a double integral?

I know how to change the intervals of an integral, for example the integral of $(\sin x)^2$ from $-\pi$ to $\pi$ is equal to $\pi\int_{-1}^1 (\sin πx)^2 \,dx$. I find it difficult to do that in 2D. In particular i want the $$ \int_1^2 \int_3^4…
Marios Theodoulou
  • 31
3
votes
0 answers

Multivariable integral over a simplex

Let $p$ be a positive integer, let $B > A >0$ and let $\beta >0 $ and $\beta \neq 1$. With a help of Mathematica (ie using elementary integration and consecutive simplifications) I have shown that : \begin{eqnarray} &&I^{(A,B)}_p…
Przemo
  • 11,331
3
votes
1 answer

How to compute the integral $\frac{1}{2\pi i} \int^\infty_{-\infty} \frac{e^{ixy}}{y - i} \, dy$

I am reading an essay that says that it is true that \begin{equation} \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{e^{ixy}}{y - i} \, dy = \begin{cases} e^{-x} & \text{for x }>0 \\ 0 & \text{for x} < 0 \end{cases} \end{equation} I…
harlekin
  • 8,740
3
votes
3 answers

Evaluate $\int_0^\infty \frac x {(x+1)^3}$

I feel like this should be easy, but I can't remember the technique that I should use to solve this. How does one solve an integral like this: $$\int_0^\infty \frac x {(x+1)^3}$$
tmsimont
  • 179
3
votes
2 answers

Solving $\int_{-\infty}^{+\infty}e^{-2\alpha|x|}\cos^2(x)\,dx$

I'd like some help solving the integral $$ \int_{-\infty}^{\infty} e^{-2 \, \alpha \, |x|} \cdot \cos^2(x) \; \, dx $$ with $\alpha > 0$ I just assumed 'integration-by-parts' was the way to go, but the first part of the product alone…
colorfly
  • 33
3
votes
3 answers

$x e ^{-\frac{x^2}{2}}$ at $\infty$ to $-\infty$

I want to know how to explain $\left. \left(x e ^{-\frac{x^2}{2}} \right) \right|_{- \infty} ^{\infty}$ is zero? Is it because the speed of exponentiation is greater than that of linear? How to prove? If using Bernoulli's rule, is there any other…
Wei Zhong
  • 457
3
votes
2 answers

Why does integration of acceleration data create a slope?

I created a 100hz sine wave in code. When I graph the waveform I get this: When I do an integration on this pure sine wave to get a velocity waveform I get: Is this normal? I do not have a math background but I am trying to understand this.
RobC
  • 135
3
votes
4 answers

Evaluating the indefinite integral $ \int 4x \sqrt{1 - x^4} \,dx$

I need help evaluating $$\int 4x \sqrt{1 - x^4} dx$$ What I have tried so far: Rewriting the integral as $$\int \frac{4x}{\sqrt{1 - x^4}} (1 - x^4) dx$$ $$\int \frac{4x}{\sqrt{1 - x^4}}dx - \int \frac{4x^5}{\sqrt{1 - x^4}} dx$$ The first integral I…
MT_
  • 19,603
  • 9
  • 40
  • 81
3
votes
1 answer

Integral of $\int\frac1{x}\sqrt[3]{\frac{1-x}{1+x}}dx$

I could substitute $t=\sqrt[3]{\frac{1-x}{1+x}}$ and get $\int\frac{6t^3}{t^6-1}dt$, which leads to partial fractions decomposition with 6 variables. That's annoying and may lead to mistakes. Is there any other way to compute this integral?
k5f
  • 641
3
votes
1 answer

How to get reduction formula of $u_n=\int\frac{x^n}{\sqrt{ax^2+2bx+c}}$?

How to get reduction formula of $$u_n=\int\frac{x^n}{\sqrt{ax^2+2bx+c}}$$ My try: Here $P_{n-1}(x)$ is a polynomial of degree…
RE60K
  • 17,716
3
votes
1 answer

Integral of exp(a/x)

I know the integral of exp(ax) but couldn't find a solution for integral exp(a/x). I very appreciate if somebody help me. Thanks Reza
reza
  • 31
3
votes
2 answers

How to give the order error term in Laplace asymptotic methods about integral?

I don't know why the order of error term is $O(n^{-1})$ or more high in asymptotically computing integral using Laplace's method? For instance, the following examples: Suppose that $h(\theta)$ is a real function, has a unique manimum at…
jameses
3
votes
2 answers

$\frac{d^2x}{dt^2} =3.$ Separation of variables

Can someone show the steps to desperate the values and do the integration for the following: $$\frac{d^2x}{dt^2} =3\quad?$$ What I have is $$d^2x=3dt^2$$ Integrate $$dx=3t^2 + c$$ $$x=t^3+ct +d$$ Is that right?
Tom
  • 31
3
votes
1 answer

$\int \frac {1}{(x^2+R^2)^{3/2}}dx $

integrate $$\int \frac {1}{(x^2+R^2)^{3/2}}dx $$ This came up doing a physics task, but I have no idea how to integrate it without straight using integration table. I tried to do it integrating by parts but that get me nowhere.
Lugi
  • 1,319
3
votes
3 answers

The integral $\int_{\varepsilon}^1 r^n(1-r)^{k-n}\,dr $

We know from this answer that for $0\leq n \leq k$, $$ \int_0^1 r^n(1-r)^{k-n}\,dr = \frac{1}{(k+1)\dbinom k n}. $$ In my case, $n$ and $k$ are integers. But what is $$ \int_{\varepsilon}^1 r^n(1-r)^{k-n}\,dr $$ for small $0 < \varepsilon$ ? This…
user66307
1 2 3
…
99 100