Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n),$ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
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A limit using integral to compute

Is it possible to compute the following limit by using integral? $$\lim_{n‎ ‎\rightarrow ‎\infty} ‎‎\left( ‎‎\dfrac{1}{n^2+1}+‎\dfrac{2}{n^2+2}+‎\cdots ‎+‎\dfrac{n}{n^2+n}‎‎‎\right)$$
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Proof that $\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4$?

Can one prove that $$\int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4?$$ I would prefer using the methods of contour integration.
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Infinite intersection of monotonic Jordan measurable sets is also measurable

I want to show that if $E_1 \supset E_2 \supset....$ is a monotonic sequence of Jordan measurable sets, so $$Z = \bigcap_{k=1}^\infty E_k$$ is of Lebesgue measure null, then Z is also Jordan ...
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Evaluate this integral without knowing the function

I'm an high-speed aerodynamics student but have some problems with some math expressions. We know that: $$\int_0^c\dfrac{\partial z_e(x)}{\partial x}dx=z_e(c)-z_e(0)$$ I'm having trouble to do such ...
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How to show that $\int_0^{\infty} dx \frac{\log{x}}{1+x^2}$ is zero using complex analysis

I want to show this using contour integration, the appropriate contour is a keyhole I think.
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Evaluating $\int_{-\pi}^{\pi} (e^{ix} + e^{-ix})^n dx$

In an exercise following identity is used: $$\int_{-\pi}^{\pi} (e^{ix} + e^{-ix})^n dx = \begin{cases} 0, \hspace{2.1cm} n = 2m+1 \\ 2\pi {2m \choose m}, \hspace{1cm} n=2m. \end{cases},$$ Does ...
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Integrability of $f\left(\frac1n\right)=\frac1n$ and else $f(x)=-1$

Let $f:[0,1]\to\mathbb{R}$ be defined as $f(x)=\cos(x)$ if $x=\frac1n$ for a natural number $n\ge1$ and $f(x)=-1$ else. Does $$\int_0^1f$$ exist? I think it is because the problematic region around ...
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Volume bounded between an Ellipsoid and a Cone?

I'm a bit confused about how I would be able to find the volume bounded by a cone of known theta and an oblate spheroid of b = c. I'm trying to use triple integrals for the solution, and I think I ...
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Calculate the limit $\lim_{x\to\infty} \int_{0}^{1} g(xz) dz$.

let $g(x)$ be a continuous function s.t. for each $x \ge 0$, $\lim_{x\to\infty} g(x)=L \ne 0$. calculate the limit : $$\lim_{x\to\infty} \int_{0}^{1} g(xz) dz$$ SOLUTION ATTEMPT: I'm thinking ...
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Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a smooth function. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ $$h(w) = \int_{S^2} d\Omega(\hat{n})\,f(\sqrt{|w|}\, \hat{n})$$ where d\... 0answers 36 views The Deconvolution Integral The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, Y(j\omega) = X(j\omega)\cdot H(... 2answers 55 views How to integrate something already having integration. The textbook wrote, \begin{align} g_n(t)&=\int_0^tf(t-s)g_{n-1}(s)ds, &n=2, 3, \cdots, &&0 \le t \le \infty \\ G_n(t)&=\int_0^tF(t-s)dG_{n-1}(s), &n=2, 3, \cdots, &&0 \... 1answer 71 views How to integrate f'(\ln x)/x given a table of values of f and f'? My professor gave me this question without any description.\int_1^e \frac{f'(\ln x)}{x}\,dx$$where f has the following data: \begin{array}{r|lllll} x& 0 & 1 &\frac{\pi}{2}... 2answers 175 views how to calculate \int_{0}^{\infty}\frac{x}{\sqrt{e^x-1}}\mathrm{d}x I was trying to solve another integral when then I reached this, I've no idea of how to select the contour for the integration. 0answers 20 views A double integral with parameters Please help me solve this integral:$$\frac{1}{\pi}\int\int_{-tz \in \left[ 0,\frac{\pi}{2} \right]} \cos(tz)\left[ (x-z)\left( e^{-t^{\alpha_1}}-e^{-t^{\alpha_2}} \right) \right] \, dt \, dz$$... 4answers 76 views If \int_{-x}^{x}f(t)dt=x^3-x^2+x+1, then find f(-2)+f(2). If$$\int_{-x}^{x}f(t)dt=x^3-x^2+x+1,$$then how can I find f(-2)+f(2)? I tried to use the derivative of integral but I get f(2)-f(-2)=9. 1answer 46 views Evaluating \int \frac{2\ dx}{\sin x \cos x}. [duplicate] Evaluate the following$$\int \frac{2\ dx}{\sin x \cos x}.$$2answers 34 views integral of a square root function by substitution. A practice problem:$$\int \sqrt{x^2+9}\ dx $$So what I did was to substitute x with \tan \theta, which yields$$\int \sqrt{9\tan^2\theta+9}\ dx $$Then I brought the 9 out$$\int ... 0answers 20 views Riesz potential of a set and its complement LetF\subset [0,1]$be a closed set,$G = [0,1]\setminus F$,$\alpha \in(1,2)$. Is there a simple condition on$F$under which the integral $$\int_F\int_G \frac{dx\,dy}{|x-y|^{\alpha}}$$ is finite? ... 2answers 56 views Reverse Power Rule integration. Ok, so I am confused about the following; When we have a polynomial, say$P(x)$, and we want to solve an integral where$P(x)$is raised to a certain power, for example; $$\int (P(x))^adx$$Why can we ... 2answers 114 views Can we interchange the limit and integration for$\int f_n(x)\,\mathrm{d}x$? [duplicate] In connection with this question about computing integrals of the form $$\mathcal{I}_n=\int_0^\infty f_n(x)\,\mathrm{d}x=\int_0^\infty \frac{\mathrm{d}x}{e^x+x^n}$$ I noticed an interesting trend ... 0answers 20 views Integral with Simpson's method not converging I'm trying to use Simpson's rule to integrate the following function in a program: $$\int_{z_a}^{z_b}\frac{Cf(z)}{(C^2 - f(z)^2)^{3/2}}\,dz$$ where$C$is a constant and$f(z)$are interpolated ... 0answers 67 views Integral of a Gaussian with Trigonometric functions Involved I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty [\sin(... 0answers 57 views Is it possible to perform Integration in this equation? I have been working on a problem for a long time and have finally arrived at this differential equation. The problem is simple, which surfaces obey the Reflection Property. Now there are several ... 3answers 81 views Find \int x\ln(x^2e^{x^2})\,dx. How can I find the following integral$$\int x\ln(x^2e^{x^2})\,dx.$$Which substitution may I use to solve the integral? 1answer 22 views Chasle relation with arbitrary ordering of endpoints. Consider the Chasle relation :$$\int_a^c f(x) dx= \int_a^b f(x) dx+ \int_b^c f(x) dx\,$$Is this only true for a\leq b\leq c or is it true for any ordering of a,b and c ? In the case a\leq b\... 0answers 33 views 2-Dimensions Integral convergence Does the following integral converge on \mathbb{R}^2: \int \int \frac{log(x^2+y^2)}{x^2+y^2}dxdy I found that inside the unit circle the integral is -\infty and outside the unit circle its +\... 2answers 33 views \sum_{k=1}^{n} \frac{1}{k} \geq \int_1^{n+1} \frac{1}{x} dx Inequality$$\sum_{k=1}^{n} \frac{1}{k} \geq \int_1^{n+1} \frac{1}{x} dx$$I don't see how you reach this inequality, or rather why it is correct. The context of this problem was the following: Show that$$... 3answers 68 views Integrate$\int \frac{x+3}{2x^2+x+3}\ dx$[closed] $$\int \frac{x+3}{2x^2+x+3}\ dx$$ How should I approach this? 1answer 90 views Double integration problem, how to integrate$e^{x^2}\$?

What is the value of $$\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^ye^{-\frac{1}{2}\left(x^2+y^2\right)}dxdy$$ I draw the region of integration tried to change the order but still i don't know ...