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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-1
votes
0answers
38 views

Integral: $\int \frac{1}{1+x^4}dx$ [duplicate]

I asked my teacher how to do it, he answered I had to use series but I won't learn this method in highschool. But I want to know how to solve it. I read online about series but I don't see how ...
1
vote
3answers
53 views

I've got the following differential equation, how do I integrate the expression to get the answer?

-Solve the differential equation ,with the given condition: $${\partial z \over \partial x}+(2e^x-y){\partial z \over \partial y}=0.\ \ z=y\ \ \ at \ \ \ \ \ x=0. $$ I solve it as follows: $${dx ...
2
votes
2answers
75 views

Why are we allowed to make trig substitutions when solving integrals?

I was taught that integrals involving $$\sqrt {a^2-x^2} \qquad \sqrt {a^2+x^2} \qquad \sqrt {x^2-a^2}$$ where $a$ is a constant can be solved by substituting various trig functions for $x$, allowing ...
1
vote
4answers
101 views

The integral of Gaussian function of three variables [on hold]

How do I solve this $$\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty}\displaystyle\int_{-\infty} ^{\infty} e^{-x^2-y^2-z^2}\ dx \ dy \ dz$$
4
votes
1answer
58 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
-1
votes
0answers
39 views

How to link these two equations? Any ideas? [on hold]

How do I link these two equations? ∫▒〖x^n e^ax dx〗 ∭_D▒〖f(r,θ,z)r dr dθ dz〗
2
votes
1answer
58 views

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
4
votes
2answers
68 views

Show that $ \int_{-\infty}^{\infty} \frac{x^3}{(x^2+4)(x^2+1)}\, dx$ does not converge

I noticed that $\displaystyle \int_{-a}^{b} \frac{x^3}{(x^2+4)(x^2+1)}$ will converge to $0$ whenever $a=b$ and will converge to some value whenever $a,b$ are in the reals (excluding infinity). How ...
1
vote
1answer
28 views

The proof of differentiating the Riemann integral

Does anyone know how to prove this: Suppose $f(x,t)$ and $\frac{\partial}{\partial t} f(x,t)$ are continuous functions on $M=\langle a,b\rangle \times \langle c,d \rangle$. Then $I$ is differentiable ...
1
vote
3answers
44 views

Integration of complex functions with trig functions: $\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$

$\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$ How should I integrate this? Using the exponantial identities of trig? Any hints will be great... Thank you!
0
votes
0answers
9 views

Parametric integral,lebesgue

can anyone help me with finding the function f(x,t) that is lebesgue integrable, bounded on a closed interval [a,b] and its true, that $\frac{d}{dt}\int_{a}^{b} f(t,x)dx = \int_{a}^{b} ...
0
votes
1answer
59 views

Integrate $\int\limits_0^\pi \mathrm{d}x \ e^{\mathrm{i}x} \sqrt{1 - e^{2\mathrm{i}x}}$

I'm trying to integrate $$ \int\limits_0^\pi \mathrm{d}x \ e^{\mathrm{i}x} \sqrt{1 - e^{2\mathrm{i}x}} $$ I tried using various trigonometric identities but nothing seems to work. A small hint would ...
0
votes
1answer
29 views

Trigonometric Integrals using residue

I need to calculate, for $a \geq 1$,$$\int_{-\infty}^\infty \frac{e^{iax}sin(x)}{x^2+1}dx$$ Attempt at Solution: $$Let \ \ f(z)=\frac{e^{iaz}e^{iz}}{z^2+1}=\frac{e^{iz(a+1)}}{z^2+1}$$ This has poles ...
2
votes
2answers
46 views

Large $a$ asymptotics of $\int_0^{\pi/4} \exp(-a(x^2-\frac{x^4}{3}))$

I'm looking for a way to prove that $\displaystyle \int_0^{\pi/4} \exp(-a(x^2-\frac{x^4}{3}))dx=\int_0^{\pi/4} \exp(-ax^2)dx+o\left(\int_0^{\pi/4} \exp(-ax^2)dx\right)$ as $a$ goes to $\infty$ ...
15
votes
2answers
337 views

Improper Integral $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral for every $n\in\mathbb{N}$? $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$$ By Mathematica we see that $$\int_0^\frac{1}{2}x\cot(\pi ...
2
votes
1answer
50 views

Calculate $\int_{-T}^T {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

Let $\lambda$ and $\nu$ be real numbers. Then, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
0
votes
0answers
9 views

Building an integral model to estimate variable based on population — where to start?

I'm trying to build a model that provides an estimate of cumulative pollution output based on the population level of a city. The pollution level for this city increases exponentially with ...
-2
votes
2answers
38 views

Integral with logarithm is positive

Given the following integral: $$I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx,$$ where we assume $\int_{\mathbb{R}} f(x)\, dx =1$ and $f\geq 0$ a.e. Assume for ...
1
vote
1answer
7 views

When $f \mapsto \lambda\int_{\mu}^{x}f(t)dx$ is contration map

$$\Phi\colon C[a,b] \to C[a,b], f \mapsto \lambda\int_{\mu}^{x}f(t)dt$$ I want to find $\lambda,\mu$ such $\Phi$ is contraction map, so $$|\lambda\int_{\mu}^{x}f(t)dt| < q|f(x)|$$ on $[a,b]$ for ...
0
votes
0answers
38 views

Finite region enclosed by two curves when $f(x)$ and $g(x)$ contain variable a

I need to find the area enclosed by the curves $f(x) = x^3 - ax$ and $g(x) = (a-1)x^2$. $a > 0 $ This I can do by subtracting the integral of the "lower function" from the integral of the "upper ...
31
votes
3answers
649 views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...
3
votes
1answer
20 views

Problem about sequence of Riemann integrable function

a) Suppose that $g_n\ge0$ is a sequence of integrable function and $\int_{a}^{b} g_n(x) dx$ converges to $0$. If $f$ is an integrable function on $[a,b]$, then $\int_{a}^{b} f(x)g_n(x) dx$ converges ...
1
vote
1answer
35 views

integrating something from a partial derivative $v=\int \frac{2x}{x^2+y^2}\,dy$

i am trying to learn harmonic analysis, and i have$$\frac{\partial u}{\partial x}=\frac{2x}{x^2+y^2}=\frac{\partial v}{\partial y}$$ and i want to get $v$. so what i do is: $$v=\int ...
1
vote
1answer
134 views

How to integrate $\int_{-\infty}^{\infty} \frac{dx}{1+x^{12}}$using partial fractions [closed]

How do I integrate the improper integral $$\int_{-\infty}^{\infty} \frac{dx}{(1+x^{12})}$$ using partial fraction decomposition? I am restricted to use only the principles taught in Calculus 2, ...
3
votes
1answer
45 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
0
votes
1answer
23 views

Evaluate the integral using CIF

Evaluate $$ I=\int_C \frac{dz}{z(z-1)(z-2)}$$ where $C = \{z \in \mathbb C: |z| = r\}, r \gt 0$. I have splitted the integrals using partial fractions and applied Cauchys integral formula and found ...
2
votes
1answer
54 views

Weird (?): Use Cauchy's Integral formula to calculate $\int _{|z|=3} \frac{z}{z^2-\pi^2}dz$

Weird (?): Use Cauchy's Integral formula to calculate $\large \int _{|z|=3} \frac{z}{z^2-\pi^2}dz$. But the function $\frac{z}{z^2-\pi^2}$ is holomorphic on all of $|z|=3$. Am I missing something? ...
0
votes
2answers
15 views

Solution verification: Integral of the conjugate equals the conjugate of the integral

For all $f(z)$ holomorphic in all $\mathbb C$ and every smooth curve $\gamma$, is it correct that: $\overline {\int_\gamma f(z)dz}=\int_\gamma \overline { f(z)}dz$ My solution: If $f$ is holomorphic ...
1
vote
0answers
31 views

Can summation $\sum_{n=[-N\ldots N]}n e^{-\frac{(y-cn)^2}{2}}$ be lower bouned by integration $\int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} dx $

I was wondering if the following summation \begin{align*} \sum_{n=[-N \ldots N]}n e^{-\frac{(y-cn)^2}{2}} \end{align*} can be lower bounded by integral $$ \int_{-N}^Nx e^{-\frac{(y-cx)^2}{2}} \, dx ...
2
votes
1answer
64 views

Arc length of natural log function

I am currently trying to find the arc length of $f(x)=ln(x)$, which involves the integral $$\int \sqrt{1+\frac1{x^2}}dx$$ I managed to solve the integral correctly but I want to know if there is a ...
6
votes
3answers
305 views

Evaluate :$\int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx$

How to evaluate $$ \int \frac{\sin^{-1} \sqrt{x} -\cos^{-1} \sqrt{x}}{\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}} dx $$ I know that $\sin^{-1} \sqrt{x} +\cos^{-1} \sqrt{x}=\frac{\pi}{2}$ but after that ...
0
votes
0answers
32 views

Definite integral of absolute value is zero iff . . .

Given $f$ is bounded on nondegenerate interval $[a,b]$. I need to prove : If $f$ is continuous on $[a,b]$, then $\int_{a}^{b} |f|=0$ if and only if $f=0$ for all $x$ in $[a,b]$. The left direction ...
1
vote
0answers
39 views

Finding probabilities of a continuous random variable

I have the following continuous random variable density function: $$ f(x) = \begin{cases} \frac14 & if\,0\le x<1 \\ \frac12 & if\,1\le x<2 \\ a & if\,2\le x<4 \\ 0 & ...
8
votes
1answer
2k views

What are some difficult integrals done by substitution and elementary functions?

What are some examples of difficult integrals that are done using substitutions? For example: $$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$$ Please no laplace and fourier transforms as I ...
-4
votes
0answers
37 views

can get help ? i need the solution for this equation [closed]

Please, I need the solution for this equation, I hope you can help to solve this equation: $$Re\int_0^1(1+x^2+ix(1-x))^{\frac56}\text dx$$
1
vote
2answers
52 views

An example of discontinuous integrable function

"Let $f(x)=1$ if $x=1,1/2,1/3,1/4,...$ and $f(x)=0$ elsewhere. Prove that $f$ is integrable on $[0,1]$. What is the value of that integral?" I'm guessing the value to be $0$, intuitively. I know the ...
29
votes
4answers
1k views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
1
vote
1answer
68 views

How to find the anti-derivative is $\left|x\right|$?

Let the function $f(x) = \cases{1 & x>0 \\ -1 & x<0}$. I want to show that the anti-derivative of $f(x)$ is $\left|x\right|$. Let's split to cases: If $x\le 0$ then $F(x) = \int_{-1}^x ...
4
votes
3answers
61 views

The shortest way to prove that $\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx $ converges.

I'm trying to show that the integral $$\int_1^\infty \frac{{\arctan \left( x \right)}}{{\sqrt {{x^4} - 1} }}dx \quad \text{is convergent}.$$ We know that $$\frac{{\arctan \left( x \right)}}{{\sqrt ...
3
votes
1answer
32 views

What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated _under_ the Koch Curve

Grew out of frustration about this question; just replace "over" by "under": What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated over the Koch Curve What is $$ \iint_{K} e^{a \cdot x + b ...
6
votes
2answers
101 views

Integration of $\dfrac{x}{\sinh x}~dx$ from $-\infty$ to $\infty$

Problem from set on recidues: Evaluate $$\int_0^\infty \frac{\log x}{(x-1) \sqrt{x}}~dx.$$ After the substitution $x = e^u$ and easy computations. The integral becomes $$\int_{-\infty}^\infty ...
6
votes
3answers
172 views

Approximate solution of differential equation

My task: find approximate solution as $$y = y_0(x) + y_1(x)\lambda + y_2(x)\lambda^2 + y_3(x)\lambda^3$$ of differential equation $$y' = \sin x + \lambda e^y, y(0)=1-\lambda. \ \ \ \ (*)$$ My ...
0
votes
0answers
20 views

Help with a proof involving integration of difference of functions

Let $f(x,a)$ and $g(x,a)$ be two continuous functions from $[0,1]\times \mathbb{R}^+ \to \mathbb{R}$. $g(x,a)$ is decreasing and convex in $x$. $f(x,a)$ is decreasing in $x$ and increasing in $a$. ...
5
votes
4answers
99 views

Maximum value of the integral $\int_0^1e^{|t-x|}dt$ for $0 \leq x \leq 1$

Define $$f(x)=\int_0^1e^{|t-x|}dt$$ I have to find the maximum value of $f(x)$ when $0 \leq x \leq 1$. To remove the modulus, I wrote $$f(x)=\int_0^xe^{x-t}dt + \int_x^1e^{t-x}dt$$ ...
1
vote
1answer
31 views

Fourier transformations and the inversion formula

I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate ...
2
votes
2answers
26 views

Integrate a sum (geometric series) round |z| = 1

This is a question from a text book (Saff and Snider, Complex analysis for mathmatics science and engeneering, page 203). Let $$ f(z) = \sum_{k=0}^\infty (k^3/3^k)z^k $$ Evaluate $$ ...
0
votes
3answers
69 views

Taylors Series for Limits

For the equation: $$\lim_{ x\to 1} \frac{1−x+\ln x}{1+\cos(\pi x)}$$ How can you evaluate this limit using a Taylor Series for both the numerator and deminator? Would I need to create a taylor ...
0
votes
0answers
32 views

Why do I get a big error when I compute this integral with Gauss-Legendre Quadrature?

I'm using Gauss-Legendre Quadrature to solve the following integral: $\int_0^{1}x^xdx$ After I've compared the result with the MatLab vpa(int(...)) of the same integral I've noticed that the ...
1
vote
3answers
43 views

Proving an identity

Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is ...
6
votes
1answer
413 views

proof of stokes theorem

I don't understand the "idea" of the following proof, as well as some of the steps. As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...