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
4
votes
1 answer

What does this formula mean?

What is this formula about? Does it make sense? I'm serious. I like this, but feel foolish when I'm asked what it does mean ;-)
Lenne
  • 157
4
votes
5 answers

Convergence of $\int_{0}^{+\infty }\frac{e^{-x}} {1+x^2}dx$

$$\int_{0}^{+\infty }\frac{e^{-x}} {1+x^{2}}dx$$ One more task that i got on my exam, and i failed. I tried with partial integration. I need to find out does it converge?
salesh
  • 185
4
votes
2 answers

$\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}$

$\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}$ I tried: $\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}=\int \frac{dx}{\sin x \sqrt{\sin2x\cos \alpha+\cos 2x\sin \alpha}}$,then i could not solve and changed the method. Let…
Brahmagupta
  • 4,204
4
votes
3 answers

definite integral without using complex line integral

It is well known that the following definite integral is proved by using complex line integral. $\int_{0}^{\pi} \log(\sin{x}) dx =-\pi\log{2}$ Does anyone know the calculation of above integral without using complex line integral?
user261031
  • 41
4
votes
3 answers

Does $\int_0^\infty |\cos(x^2)| \mathrm dx$ converge?

I know this is a silly question, but I've tried to find an answer using my TI-89 calculator, Maple and wolframalpha but none of those could tell me whether $$\int_0^\infty |\cos(x^2)| \mathrm dx$$ converges or diverges. Thus, I'd be very happy if…
Huy
  • 6,674
4
votes
0 answers

In integral, would it make any sense to write dx as denominator?

Does it make any sense to write something like this? $$ \int \frac 1{dx} $$ I think I saw it somewhere, but since I can not find anything similar on wikipedia and wolfram, I have to assume that this is actually nonsese. So is it? Closest thing found…
industryworker3595112
  • 165
4
votes
2 answers

Is $\int_a^a f(x) dx$ always zero?

Is the result: $$\int_a^a f(x) \,\text{d}x = 0$$ always zero? This seems obvious at first, but what if $f(x)$ diverges at $x=a$? For example, Wolfram Alpha tells me $$\int_0^0 \frac{1}{x}\,\text{d}x = 0\qquad…
Sciencertobe
  • 530
4
votes
0 answers

Computing $\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$

I am trying to compute $$\int_{0}^{1}\int_{0}^{1}\int_{\max\{x,y\}}^{1} e^{z^3} dz dx dy$$ What I have done is to reverse the order of integration; so I did the integration with respect to $x$ and $y$ first. I get this in the…
AstroInt
  • 333
4
votes
2 answers

this integral $\int\frac{2-\sin{x}}{\sin^2{x}-\sin{x}+1}dx$

Find integral $$\int\dfrac{2-\sin{x}}{\sin^2{x}-\sin{x}+1}dx$$ As suggested ,we take the Weierstrass Substitution is often useful in integrals such as the…
user223800
4
votes
3 answers

Double Integrals involving infinity

Double Integrals involving infinity $$ \int_0^\infty\int_0^\infty xye^{-(x^2+y^2)}\,dx\,dy $$ my work let $$t = x^2 +y^2$$ then $$ dt =2x\,dx$$ since $$ye^{-t}$$ $$[ye^{-t}]$$ from $\infty$ to $0$ what next stuck here
user155971
  • 1,515
4
votes
4 answers

Find $\int_{-\infty}^{\infty}xe^{-x^2/2}dx$

$$\int_{-\infty}^{\infty}xe^{-x^2/2}dx$$ I have made an attempt at this by substituting u=x^2 to get: $$\frac12\int_{-\infty}^{\infty}e^{-u/2}du$$ This gives me: $$\frac12[-2e^{u/2}]_{-\infty}^{{\infty}}$$ Firstly is this right and secondly, how do…
RobChem
  • 909
4
votes
4 answers

Exponential integration problem

$$ \int 2^{3x} \times 5^x \times 3^{2x} dx $$ I think we're supposed to convert all the terms into log form, but I'm not sure, and other than that I have no idea how to tackle this problem.
GinaJay
  • 147
4
votes
3 answers

Problem with Integral attempt

Problem: Evaluate: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-1}(\pi x) - \tan^{-1}x\right)dx.$$ Attempt: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-1}(\pi x) -…
User1234
  • 3,958
4
votes
3 answers

Arc Length of $x^{\frac{2}{3}}+y^{\frac23}=1$

How do I find the arch length of $$x^{\frac{2}{3}}+y^{\frac23}=1$$ The hint given was "4x the arc in first quadrant" I think I am supposed to use the formula: $$L=\int^b_a \sqrt{1+(f'(x))^2} dx$$ I tried plotting the equation in a graphing…
Jiew Meng
  • 4,593
4
votes
1 answer

Double integral (and for the area enclosed by a lemniscate).

a) Transform into polar coordinates and compute the integral $\int_\Omega ln(1+x^2+y^2)d(x,y)$ where $\Omega$ is the interior of the unit circle in the first quadrant. b) $\Omega\subseteq R^2$ which is the subset enclosed by the $Lemniscate$…
Konstantin
  • 199
1 2 3
…
99 100