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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Find $\int (x^3+1)(\cos(2x)) dx$

What I've tried for this problem is expanding it to $x^3\cos(2x) + \cos(2x)$ and then evaluating the respective functions as separate integrals. The first one uses tabular and the second one is simple u substitution. Is my procedure correct?
user159778
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Integration by parts polynomial times logarithim

$$\int \sqrt{x}\log_2(x) \, dx$$ Integral of Square root of x times log base 2 of x dx.
user159778
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erf-type integral involving sin

A buddy and I are hung up on this integral. Prove that: $\displaystyle \int_{0}^{\infty}e^{-ax^{2}}…
Cody
  • 14,054
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Calculating a ``density of states''

Let $n>0$ and $T > n$ be integers and let ${\bf c}$ be a symmetric, real and positive definite matrix. We define the ''density of states'' of a sample covariance matrix. In other words we have: \begin{equation} \omega^{(n,T)}({\bf c}) :=…
Przemo
  • 11,331
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How find this integral $\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$

let $$D=\{(x,y)|y\ge x^3,y\le 1,x\ge -1\}$$ Find the integral $$I=\dfrac{1}{2}\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$$ My…
math110
  • 93,304
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computing integral without softwares: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$

I was wondering if this integral can be solve without wolfram and others: $$\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$$ Thanks.
abii
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Liner integral?

Can someone simply explain to me how to calculate linear integral linke below? $$\int_{L} 5y \mathrm{d}L$$ Where L is line segment from (0;0) to (0,2;0,2).
Miko Kronn
  • 121
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How to simplify this double integral using substitution?

I want to take this integral: $$\int_{0}^{L}\int_{0}^{L} |x-y| \,dx\,dy$$ And set $u = |x-y|$ and convert this integral into something that only needs one integral. I don't know the correct way to do this. I can't just go: $$\int_{0}^{L}\int_{0}^{L}…
user176201
  • 35
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Solve integral $\int \sqrt{3x^2 - 2x}\ dx$

Find the following: $$\int \sqrt{3x^2 - 2x}\ dx$$ I've tried completing the square and doing trigonometric substitution but I think I am making a mistake somewhere. Thanks!
Melanie
  • 31
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Verifying Green`s theorem where boundaries are semicircle

I have problem with verifying Green`s theorem for integral: $$\int _C (x^2+y^2+\cos(x))dx+(x^2+y^2+\sin(y))dy$$ C is the boundary of the semicircle: $$\{(x,y) \in R^2: x^2+y^2\leqslant4 \wedge x\geqslant0\}$$
Robert
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Integrating with respect to time a double derivative $\ddot\phi + \frac bm\dot\phi = \frac{F}{mr}$

$$\ddot\phi + \frac bm\dot\phi = \frac{F}{mr}$$ I want to integrate this to get an equation for $\dot\phi(t)$ but I don't know how to integrate double derivatives. The answers say that the homogeneous equation associated with it is: $$\ddot\phi +…
user135688
  • 635
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Integration of this type?

Can anyone help me with this? I used completing the square but do not know how to continue? Thanks 9.. Gaussian Integral The following definite integration is particularly relevant in the subject of quantum mechanics (QM). (It always pops out, so…
Lim Zhi Jian
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Limits of an Integral: $\lim_{n \to \infty}\int f(x)\sin(nx)dx=0$

I have a problem with the following exercise. I don't really have an idea where to start. I'm glad about every help. So here is the exercise: Suppose $f\colon \Bbb{R} \to \Bbb{R}$ is a function in $L^1(\Bbb{R})$ (i.e integrable function). Show…
swissmathegirl
  • 31
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Antiderivative of $\exp(x^2)$

Can you please provide a step by step solution for next integral. I don't have any idea of how this can be solved $\displaystyle\int e^{x^2}\,dx$.
Jacob
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Indefinite integral of $\frac{\ln(x)}{(x-3)^2}$

I am trying to compute the integral $$ \int\frac{\ln(x)}{(x-3)^2}\mathrm{d}x $$ I have tried the following substitution, but seem to get nowhere: $u = x - 3$. $$x=u+3$$ $$dx=du$$ $$\int\frac{\ln(u+3)}{u^3}\mathrm{d}u$$ I get stuck at this point and…
JoshBoiskin
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