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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0
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2answers
17 views

Finding a function satisfying two tangents?

I want to find a function $f$ that satisfies the following conditions: (i) $f'(x)=x^3, \forall$ $x\in \mathbb R $. (ii) $x+y=0$ is a tangent to the graph of $f$. If we integrate (i), we get ...
0
votes
1answer
30 views

Indefinite INTEGRAL fraction [duplicate]

Compute the indefinite integral Irrational: $$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$ Help me pls. What do I need to do then? fraction isn't simplified enter image description here
-1
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0answers
16 views

Can a curve be represented by a differential equations? [on hold]

How from the variational principle curvature shape of a free hanging chain held at two fixed ends.. ?
2
votes
1answer
23 views

Determine the integral $\oint_{|z|=1} (\frac{1}{z}+\frac{2}{z^2}) dz$

Determine the integral $\oint_{|z|=1} (\frac{1}{z}+\frac{2}{z^2}) ~dz$. My answer: \begin{align*} \oint_{|z|=1} (\frac{1}{z}+\frac{2}{z^2})~ dz&=\int_0^{2\pi} ...
0
votes
1answer
31 views

Integral equation: $x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t$ [on hold]

Would you please find the function $f$ such that $$x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t \quad ?$$ Thank you.
0
votes
0answers
9 views

Existence of weak derivative of a translation of a function

Let $f$ a function in $L^1_{loc}(\mathbb{R^n})$ such that his weak derivative of order $\alpha$, $D^{\alpha}_wf$, exists. We consider a vector $h\in \mathbb{R^n}$ and we define $g(x)=f(x-h)$. I have ...
0
votes
1answer
32 views

Characterization of elements of $X^*$ via the Radon-Nikodym theorem

I am reading Lindenstrauss' Classical Banach Spaces II and I am having trouble with the following characterization of integrals. First a couple of preliminary definitions: Let $(\Omega, \Sigma, ...
7
votes
2answers
80 views

An integral related with the Riemann $\zeta$ function

I have to prove that: $$ \forall s>1,\qquad\int_0^\infty \sum_{k=1}^{\infty}\frac{1}{(k^s+1)^x+k^s}dx=\zeta(s). $$ I how do I find the closed form for this sum? $$ ...
-3
votes
0answers
40 views

Indefinite INTEGRAL fraction [on hold]

Compute the indefinite integral Irrational: $$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$ multiplying by conjugate enter image description here Help me pls. What do I need to do then? fraction isn't ...
0
votes
0answers
9 views

Preserving the positive correlations of two functions with a mixed measure

Let $V_1,V_2:\mathbb{R}\rightarrow \mathbb{R}$, be two convex functions which admit a minimum (which implies $\lim_{x\rightarrow \pm\infty}V_i(x)= \infty $). Let ...
3
votes
2answers
146 views

How to evaluate $\int_0^\infty \frac{e^{-x}+x-1}{x(e^{2x}-e^{-2x})}dx$? [on hold]

We are unable to verify this this equality $$ 4\int\limits_0^\infty \frac{e^{-x}+x-1}{x\left(e^{2x}-e^{-2x}\right)}\;\mathrm{d}x=\gamma+\ln\frac{16\pi^2}{\Gamma^4\left(\frac{1}{4}\right)}\;. $$ ...
1
vote
2answers
56 views

Evaluate the improper integral $\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$.

$$\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$$ I've tried hard for this but of no use.I've applied integration by parts by which I get $$\int_0^\infty \exp(-sk)\sin(kx)\,dk=\frac{x}{x^2+ s^2}.$$ ...
0
votes
2answers
25 views

If f*g is Riemann integrable, g continuous, nonzero and bounded, show that f is Riemann integrable

How would I go about proving that if $fg$ is Riemann integrable, given that $g$ is continuous, nonzero, and bounded (so $g$ Riemann integrable), how would I go about showing that $f$ is Riemann ...
2
votes
1answer
76 views

Proving Fatou type lemma

Let $f_1, f_2, \cdots$ and $f$ be nonnegative lebesgue integrable functions on $\mathbb{R}$ such that $$\lim_{n \to \infty}\int_{-\infty}^y f_n(x)dx = \int_{-\infty}^y f(x)dx \; \; \text{ for each ...
0
votes
1answer
29 views

For every integrable $f: (X,\mu) \to \mathbb{R}^n$, $||\int f d\mu|| \le \int ||f||d\mu$.

For every integrable $f: (X,\mu) \to \mathbb{R}^n$, $||\int f d\mu|| \le \int ||f||d\mu$, where $||\cdot ||$ is the standard Euclidean norm. Expanding the first one, I get $\sqrt{\int f_1^2 + \cdots ...
1
vote
0answers
6 views

Nontrivial integral involving Hilbert Transform

I am trying to compute the following integral: $$\int \mathbb{H}(y_x(s)) \ ds$$ where $$\mathbb{H}(F)=PV\int_{-\infty}^{\infty} \frac{F(x')}{x-x'}\ dx'$$ is the Hilbert transform, and $PV$ means we ...
0
votes
0answers
11 views

Evaluation of an integral associated with integral kernel of resolvent of Laplacian

I came across evaluating the following sort of integral when I was considering the integral kernel for resolvent of Laplacian $(I-\Delta)^{-1}$: $$ ...
1
vote
1answer
15 views

Surface integral of function over intersection between plane and unit sphere

I've been asked to compute the integral of $f(x, y, z)= 1 - x^2 - y^2 - z^2$ over the surface of the plane $x + y + z = t$ cut off by the sphere $x^2 + y^2 + z^2 = 1$ for $t \leq \sqrt3$ and prove it ...
1
vote
1answer
16 views

Trouble with parametrizing function

hope you're all well! I just started learning about line integrals in class today, and I'm having a difficult time understanding how and why the solution manual came up with the parameterization for ...
0
votes
0answers
10 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
-1
votes
0answers
23 views

Finding antiderivative of h

If $h(x) = -x^2$ for $x<0$ and $h(x) = x+2$ for $x \geq 0$, find an antiderative of h if it exists. I cannot find the Darboux's Theorem for integrability anywhere in my notes and I can't remember ...
-4
votes
1answer
28 views

Solve the DE: $\dfrac{ dy}{dx} =\dfrac{y(x+y)}{x(y-x)}$ [closed]

I worked it out and my answer came to be $y^3=x^3+3K$. Is my answer correct since I don't have any answer to this question. Thanks
2
votes
1answer
41 views

If Darboux (Riemann equivalent), then Lebesgue?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be compact, define $$D^+(f):= \inf\left\{\int t:t\geq f, t= \text{step function}\right\}$$ $$D^-(f):= \sup\left\{\int t:t\leq f, t= \text{step ...
1
vote
2answers
21 views

Fubini's Thm used to evaluate Integrand

$$\int_0^1 \left(\int_y^1 sin\frac{y}{x}dx\right) dy$$ I was working on some new material and came across this. Honestly, I feel as though this is just going over my head and feel a bit dumb for not ...
2
votes
1answer
30 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
1
vote
1answer
30 views

Find convergence domain of the integral

Find convergence domain of $$\int_0^\infty \! \frac{\cos^2{x}}{x^p} \, \mathrm{d}x$$ I've tried to use $\frac{\cos^2{x}}{x^p} < \frac{1}{x^p}$, but $\int_0^\infty \! \frac{1}{x^p} \, \mathrm{d}x$ ...
6
votes
3answers
154 views

How do I go about solving this?

I have tried substitution, but it is not working for me. $$ \int_0^\pi \frac{dx}{\sqrt{(n^2+1)}+\sin(x)+n\cos(x)}=\int_0^\pi \frac{n dx}{\sqrt{(n^2+1)}+n\sin(x)+\cos(x)}=2 $$ General form of this ...
0
votes
1answer
9 views

Inequality of integrals with respect to a signed measure and to total variation measure

I am trying to solve this exercise: Let $\mu=\mu^+-\mu^-$ be the Hahn-Jordan decomposition of a finite signed measure on a measurable space $(X, A)$. Show that for any bounded measurable ...
1
vote
1answer
22 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
0
votes
2answers
43 views

How to find $\int \frac {sinh(lnx)} {x}$

I've tried $\int \frac {sinh(lnx)} {2x} dx = \int \frac {e^{lnx}-e^{-ln{x}}} {2x} dx = \int \frac {x-e^{-ln{x}}} {2x} dx $
1
vote
3answers
54 views

Lebesgue Dominated Convergence Theorem example

For $x>0$ we have defined $$\Gamma(x):= \int_0^\infty t^{x-1}e^{-t}dt$$ Im trying to use Lebesgue's Dominated Convergence theorem to show $$\Gamma'(x):=lim_{h\rightarrow ...
-1
votes
1answer
53 views

Integral of $\int \frac{xe^x}{\sqrt{1+e^x}} dx$ [closed]

I need help to solve this: $$\int \frac{xe^x}{\sqrt{1+e^x}} dx$$
2
votes
2answers
102 views

How can $\int_a^b f(x)dx $ exist if either $f(a)$ or $f(b)$ does not exist?

In class, I came across the integral: $$\int_0^1 \frac{dx }{\sqrt{1-x^2}}=\frac{\pi}{2}$$ This is easy enough to prove using a substitution or by recalling the derivative of $\arcsin x$. However, ...
0
votes
1answer
27 views

Given two functions, if one is greater on an interval, how to prove its integral is also greater?

Working with the definition of integral as in Spivak's Calculus book, I got a big struggle to prove the next statement (even though it seems like something very obvious). Please send help. ;_; Let ...
1
vote
2answers
49 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a ...
2
votes
1answer
38 views

Solve integral weird upper bound approaching zero.

I am trying to solve an integral of the form $$p(0,t+\delta t)=\int_0^{\mu \delta t} f(x,t) dx$$ for $\delta t \rightarrow 0$ Intuitively, I would think that this integral has an upper bound which ...
2
votes
1answer
42 views

Proof this integral is equal to this sum.

I was able to determine the integral with trials and errors and we arrive with this sum, but was not able to proof it. So if anyone can proof it and also can offer a closed form. $$ \int_0^\infty ...
0
votes
2answers
39 views

$\int^{2 \pi}_0 \frac{1}{3+2 \cos t}dt$ using $\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$ or using $u=\tan \frac{t}{2}$

Question : Compute the integral of $$\int^{2 \pi}_0 \frac{1}{3+2\cos t}dt$$ I am stucked on this problem since a good while. I think we could convert that real integral into complex integral and ...
2
votes
2answers
84 views

Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques, e.g. with substitution. However, whenever I am ...
0
votes
1answer
46 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
3
votes
1answer
99 views

Is there a closed form of this integral $ \int_0^\infty \sin(xe^{-x})dx\, $? [on hold]

I have tried by subsititution method and it got more complicate than before. Can anyone help me to evaluate this integral. $$ \int_0^\infty \sin(xe^{-x})dx\,. $$
0
votes
1answer
27 views

Why most of the books give Definite Integral Represents Area Under the curve

According to my understanding The definition of Definite Integral is: if $f(x)$ is a Continuous function in $[a \:\: b]$ and if $P$ is Partition of the Interval $[a \:\: b]$ Then $$ \lim _{\lVert P ...
1
vote
1answer
26 views

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$? [closed]

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?
0
votes
3answers
60 views

$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$, $\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$ - Cauchy integral?

Compute the integrals $$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$$ and $$\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$$ I am stucked on these problems since a good while. Is there someone is able ...
1
vote
1answer
36 views

Integral and Cauchy theorem

Question : Compute the integral of $$ \int^{2 \pi}_0 \frac{1}{3+2\cos t}dt. $$ Indication: take the path $\gamma: [0,2 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$ and the integral of $$ \int_{\gamma} ...
0
votes
1answer
13 views

$\int_{\gamma} \frac{f(z)}{z^3}dz$ - Cauchy formula

Compute the integral $$\int_{\gamma} \frac{f(z)}{z^3}dz,$$ where $f(z)=az^3+bz^2+cz+d$ and $\gamma : [0, 4 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$. So by the Cauchy formula $\int_{\gamma} ...
2
votes
1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
2
votes
2answers
63 views

Does an analytical form exist for the following integral

I have an integral $$f(n,a)=\int_0^{2\pi}\mathop{\mathrm{d}x}\frac{\cos(nx)\cos^2x}{1-a\cos^2x},$$ where $n$ is an even integer and $0<a<1$ is a real number. Does an analytical form exist for ...
2
votes
1answer
43 views

Argue that the iterated integral of a continuous function is continuous

Suppose that $f : [a, b] \times [c, d] \to\mathbb R$ is a continuous function. Let $$G(y)= \int_a^b f(x, y) \, dx$$ $$H(x)= \int_a^b f(x, y) \, dy$$ Prove that $G$ is continuous on $[c, d]$ and $H$ ...
1
vote
4answers
120 views

Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$

Assume $f$ is Riemann integrable and nonnegative over $[a,b]$. Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$. This seems very obvious to me. One thing I would ...