All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
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7 views

Spectral density of a sample covariance matrix in a Gaussian Random Ensemble

Let $N > 0$ and $T > N$ be integers and $C$ be a real, symmetric $N \times N$ matrix.The question is to compute the following integral: \begin{equation} U(t) := \frac{1}{N} ...
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1answer
30 views

Integral related to a geometry problem

In the question Geometry problem involving infinite number of circles I showed that the answer could be obtained by the sum $$\sum_{k=0}^\infty \int_{B_k} \frac{4}{|1 + (x + i y)|^4} \, dx \, dy,$$ ...
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3answers
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Formula for $\int {t^n \, e^{t}}dt$?

Consider:$$\int {t^n e^{t}}dt$$ is there any closed formula for this? W|A gave me this but I don't know what is Gamma function: $$\int {t^n e^t dt} = (-t)^{-n} t^n \Gamma(n+1, -t)+ \text{constant}$$
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4answers
38 views

General form for these types of integrals

I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables
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Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
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1answer
38 views

How to integrate $\frac{1}{(x^3-a^3)^3}$ for a few limits

I need to integrate this: $$ \int\frac{1}{(x^3-a^3)^3}\mathrm{d}x $$ For a few limits: $(-a, \infty)$ and for $[0,\infty)$. Just to clarify: $$ \int_{-a + \varepsilon}^{\infty} \text{and} ...
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1answer
51 views

Alternative view on integration?

The question is about an alternative view on formulating or arriving at the concept of the integral (in case this is possible of course). Let's say we want to add a series of values $f(x_i)$ occuring ...
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1answer
33 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
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2answers
45 views

Why is the integral of any orientation form over $\mathbb{S}^1$ non zero?

I am trying to understand the proof of Theorem 17.21 in Lee's Introduction to smooth manifolds; however I am finding myself stuck right at the beginning. The statement I am having trouble with is: ...
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1answer
26 views

Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
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49 views

Evaluating the integral $\int \frac{dx}{\sqrt{a(1+x)^3+1}}$ [duplicate]

How can I solve this integral? $$\int \frac{dx}{\sqrt{a(1+x)^3+1}}$$ where $a>0$. I have tried by using Mathematica, but it fails. Someone has any sugestion?
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2answers
64 views

Evaluate $\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}dx$

Evaluate $$\int\limits_2^4\frac{\sqrt{x^2-4}}{x^2}dx$$ My working: $x=2\sec\theta\quad\Rightarrow\quad\theta=\arccos\left(\frac{2}{x}\right)$ $dx=2\sec\theta\tan\theta d\theta$ ...
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1answer
25 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
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0answers
7 views

Histogram Separation Energy Equation

I am working in level set method, specially Lankton method paper. I try to implement Histogram Separation (HS) Energy problem (Part III.C). It based on Bhattacharyya to control the evolution of ...
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1answer
18 views

Need help solving an integral for Lagrange Remainder Proof

This image of part of a proof for the Lagrange Remainder for Taylor's Formula. I need help solving the last integral. Can anyone explain?
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83 views

Integral of Sinc Function Squared Over The Real Line

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
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Simplification of a polynomial before Asymptotic series expansion

I am wondering about a very basic point related to "Asymptotic series expansions". There is a function $f(R)$ which must be expanded as $R$ goes to $ \infty $. Consider that $f(R)=g(R)*p(R)$ where ...
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1answer
52 views

Evaluate $\int\frac{\sqrt{9-x^2}}{x^2}dx$

I am trying to solve $$\int\frac{\sqrt{9-x^2}}{x^2}dx$$ My answer is slightly different to the memo: Your help is appreciated!
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5answers
75 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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114 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
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0answers
16 views

Duality relation with respect to differential operators

I have the following differential operator $$L^{\pm}h(x,y)=\pm\frac{x}{2}\frac{\partial h}{\partial x}(x,y)\pm\frac{y}{2}\frac{\partial h}{\partial y}(x,y)+\frac{1}{2}\frac{\partial^2 h}{\partial ...
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1answer
60 views

Integral $\int\frac{dx}{x\sqrt{x^2+4}}$

While solving this: $$ \begin{align} \int\frac{dx}{x\sqrt{x^2+4}} &=\int\frac{t(-1/t^2)dt}{\sqrt{(1/t)^2+4}}\tag{t=1/x}\\ &=\int\frac{(-1/t)dt}{(1/t)\sqrt{1+4t^2}}\\ ...
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1answer
113 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
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4answers
53 views

Finding the integral of a square root

Here is what I need to find $$4\int\sqrt{t^2-2}\ dt$$ Is there a way to find this without "guessing"?
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1answer
55 views

Integrals related to the function $F(x) = \int_1^x (e^t/t )\, dt$

I'm having some trouble with part of a problem from Apostol Volume 1(Section 6.26, Number 6). For completeness I'll include the whole question: A function $F$ is defined by the following indefinite ...
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4answers
95 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
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92 views

Integrate $\int \sqrt{(\sec{x} +\tan{x})}\ \cdot \sec^2x\,dx$

Integrate: $$\int \sqrt{(\sec{x} +\tan{x})}\ \cdot \sec^2x\,dx$$ My attempt : I substituted $\sec{x} + \tan{x} $ as $t^2$ Then, $$ (\sec{x} \cdot \tan{x} + \sec^2x) dx =2tdt$$ $$\sec{x}( ...
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Integration By Parts and convergence

I have an integral as $\int_0^\infty e^t w(t)f(t)dt$ where $f$ is a continuous PDF function and $w(t)\to 1$ at infinity. Is there any way to solve such an integral by parts? At the first glance ...
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1answer
19 views

Integrating a Dirac delta function with the argument dependent of a parameter

How can I handle the integral $$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$ with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.
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1answer
57 views

To be integrable, a function must be bounded; yet $\int (1/x^3)\,dx = -1/(2x^2)$?

One condition of integrability is that the function is bounded across the interval. $1/x^3$ however has a pole at $x=0$ yet it's integral is defined over all real numbers. Doesn't this violate the ...
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1answer
36 views

Step function and integration

For $f$ a continuous function on $[0,1]$ how to show that for any $\varepsilon>0$ there exists a function $$ \phi(x)=\sum_{k=1}^N \phi_j\chi_{[a_k,b_k)}(x) $$ such that $\phi(x)\geq f(x)$ for $x\in ...
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1answer
102 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
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459 views

Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...
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29 views

Countable and uncountable sets in Riemann integration

The Riemann integral over $[a,b]$ of a continuous function $f$ is $$\int\limits_a^bf(x)dx=\lim\limits_{\delta\rightarrow 0} \sum\limits_{i=0}^{n-1} (x_{i+1}-x_i)f(c_i)$$ where $c_i\in[x_i,x_{i+1}]$ ...
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1answer
42 views

Integral notation

I have encountered the following integral: $\int_{x-d}^{x+d}f(y)dy$ I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the ...
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46 views

Integration of log(sin(x)) [duplicate]

Can anyone please help me the following indefinite integral: $$\int \log(\sin(x))dx$$ Thanks
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2answers
35 views

Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
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3answers
45 views

Negative functions and their anti-derivatives

Let $f(x)$ be a continuous function with antiderivative $$F(x)=\intop f(x) \mathrm{d}x,$$ where the constant of integration is zero. Does the following hold true: ...
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Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
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1answer
34 views

calculate $\int_K \langle x,A^2 x \rangle \mathrm d x $

Let $A \in \mathbb{R}^{n \times n}$ be symmetric and invertible. $K=\{ x \in \mathbb{R}^n : \|Ax\|_2 \le 1 \}$. Now I have to calculate: $$\int_K \langle A^2x, x \rangle \mathrm d x $$ It exists a ...
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2answers
136 views

How find this integral $I=\int_{0}^{1}\int_{0}^{1}\frac{\ln{(1+xy)}}{1-xy}dxdy$

Find this integral $$I=\int_{0}^{1}\int_{0}^{1}\dfrac{\ln{(1+xy)}}{1-xy}dxdy$$ My try: since $$\dfrac{1}{1-xy}=\sum_{n=0}^{\infty}(xy)^n$$ so ...
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1answer
57 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_{0}^{\infty}\frac{e^{- x f}}{m+x}\gamma\left(a,h x\right) dx$$ where $f,m, and h \in reals$ f,m, and h are $>$ 0 $a \in integers$, a=1,2,3,... ...
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1answer
53 views

Integral of an exponent of an exponent

For a homework problem, I have to integrate this: $$\int{4^{(4+x)^x}}dx$$ How would I go around to starting this question? I don't know how to evaluate this, and I've tried to use u-subs and ...
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33 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
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1answer
44 views

Calculating an integral with a matrix

I want to calculate the following integral: Let A be a symmetric, invertible matrix. $\int_{K}<A^2x,x>dx$ where $K:=\{x\in \mathbb R^n : \|Ax\|_2\leq1\}$ A is symmetric, hence there is an ...
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1answer
108 views

A strange answer for $\int _{-1}^1 logx dx$

I typed the integral of $\int _{-1}^1 logx dx$ on wolfram alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening. Thanks.
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3answers
33 views

Finding the partial derivatives of $h(x)=\int_{0}^{\|x\|} f(t)\, dt$

Find the partial derivatives of $$h(x_1,\dots,x_n)=\int_{0}^{\|x\|} f(t) dt$$ where $\|x\|$ is the Euclidean norm of $x=(x_1,\dots,x_n)$ and $f$ is some continuous function. I'm sorry but I'm really ...
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2answers
67 views

basic question on itegration [on hold]

is $$ \int(A + B) \cdot \,dl = \int A \cdot \,dl+ \int B \cdot \,dl $$ or in others words is integral of sum of two vectors equal to sum of integral of the two vectors
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28 views

Halmos Measure Theory section 39 Theorem D

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ...