# 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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### Integrate $4x/(x^4-1)$ dx

I'm having trouble solving the integral $$\int_{5/4}^{13/12}\frac{4x}{x^4-1}\,dx$$ I have a feeling it is to do with the log integration identity but can't seem to manipulate it without involving ...
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### How would I reverse the order of these integrals?

In the question, x is the first integral at the moment. Its $\sin^{-1}(y^2)<x<\pi/2$. Y is the second integral which needs to become the first. Its $0<y<1$. I know how to integrate the ...
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### Different forms of remainder in Taylor series

In the literature, it is common to find different forms of the remainder function in a truncated Taylor series. To name a few: Integral form Lagrange form Cauchy form First, can you tell me any ...
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### Lipshitz Integral for $a=0$

I knew that this, $$\displaystyle{\int_0^\infty e^{-ax}J_0(bx)dx=\frac{1}{\sqrt{a^2+b^2}}},$$ holds for $a>0$ but, in an exercise from Arfken, it said that this holds for $a\geq0$. How can I prove ...
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### Convergence of Definite Improper Integrals of the Form $1/x$

Given a simple integral of the form: $$\int ^1 _{-1} \frac{1}{x} \, dx =\lim _{a\rightarrow 0} \int ^1 _a \frac{1}{x} \, dx + \int ^a _{-1} \frac{1}{x} \, dx$$ Is it possible to say that this ...
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### Initial value problem without explicit constant finding

So I was reading my professor's differential equation book until I came across a weird way of calculating a initial value problem without explicity having to calculate the constant that made that ...
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### Is there any integral for sin(x)/x [duplicate]

What is the integration of the function $$f(x)= \frac{\sin(x)}{x}$$ I have seen many stopping at here.. If I integrate using integration by parts it goes on..
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### Inverse Substitution ( Trigonometric Integration Rule Generalized )

I read that we can generalize the kind of substitution that is used in Trigonometric Substitution. The idea is that we replace the old variable with a new one (like we do in U-Substitution) but unlike ...
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### Problem using triple integrals and the directional derivative

Sorry for the vague question, but I'm so lost on this problem I don't even really know what it's asking of me. If anyone could show me how to do it I would appreciate it so much. All I can think of is ...
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### Can be justified $\zeta(3)=\lim_{n\to\infty}-3\sum_{k=1}^n\sum_{\nu=0}^\infty\frac{(-1)^{\nu+1}}{\nu+1}\binom{3k^3-1}{\nu}$?

My main goal is understand useful facts about my computations, that could be wrong, the way shold be too a street without exit looking for a evaluation of Apéry's constant, since I don't use any ...
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### Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$

Let $f$ be continuous function on $[a,b]\times [c,d]$. Prove that Prove that $$\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$$ First of all, note that ...
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### Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
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### integrate the improper function in (0,1)

Find $\int_{0}^{1} f(x)\mathrm{d}x$ hwere $f(x) = 2x\sin(\frac{1}{x})-\cos(\frac{1}{x})$, $x$ in $(0,1)$ 0, for all other values
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### Proof of this integral identity

I was going through some old papers on CFT particularly, 'CONFORMAL COVARIANT CORRELATION FUNCTIONS - Ferrara and Parisi' where an integral identity is used to motivate shadow operator formalism in ...
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### Green's Theorem on Line Integral

I am asked to find the line integral for the following field: $$F = (e^{y^2}-2y)i + (2xye^{y^2}+\sin(y^2))j$$ On the line segment with points $(0,0),(1,2)$ and $(3,0)$. I have to do it with Greens ...
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### Evaluating the flow rate of water across the net

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by $\vec{v}=(x-y)\vec{i}+(z+y)\vec{j}+z^2\vec{k}$ and the net is ...
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### Changing the measure

Suppose that I have the following situation: everything takes place on complex plane, I have some probabilistic measure $\mu$ and suppose that I have a family of function $(u_w)_w$ parametrised by $w$ ...
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### Volume between $z = 3\sqrt{x^{2} + y^{2}}$ and $x^{2} + (y-1)^{2} = 1$ and $z = 0$

Find the volume between $z = 3\sqrt{x^{2} + y^{2}}$ and $x^{2} + (y-1)^{2} = 1$ and $z = 0$ I am not sure how to approach finding the limits of integration. Would I need to change coordinate systems?...
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### Why change the sign of the integral when switching the limits of integration?

So I'm refreshing on integration and venture across a property of definite integrals I've been taking for granted for quite some time: \int_{a}^{b} f(x) \mathrm{d}x =-\int_{b}^{a} f(x)...
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### how to integrate the definite integral using residue theorem? [duplicate]

How to evaluate $\int_{0}^{\infty}\dfrac{1}{x^a+1}dx$, where $a>1$. I don't know where to start since $x^a+1$ could have infinitely many roots, then it is impossible(?) to evaluate its residues. ...
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### Integral of the product of a power function and an arbitrary exponentiated function

I was only able to find integral tables that solve $$f(t)=\int t^c e^{kt}dt$$ but the integral I'm trying to solve has a function, not a constant, for the exponent: $$f(t)=\int t^c e^{g(t)}dt$$ Is ...
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### A function and its derivative chasing tails

For which $t\ge0$ does there exist a differentiable function $f$ with $f(0)=0$, $f'(x)>f(x)$ for all $x>0$ and with $f'(0)=t$? This question was inspired by (and is a variation of) the ...
### Find surface area of the portion of the planar surface $S(u,v)=<u,v,1-u-v>$
Where $(u,v)$ belong to the triangle in the $uv$-plane with vertices $(0,0) (1,0) (0,1)$. I am completely stuck here!! I am sure I will use double integrals (or at least I think so) and also the ...