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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1answer
50 views

Integral $\frac{\sqrt{1+x^2}}{x}$

I understand that I will be using trig substitution, and tangent will be what is used, but I get confused later down the road when integrating with the trig.
0
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0answers
10 views

When is the riemann integral of a finite-variation process square-integrable?

Let us say $X_t$ is a continuous finite-variation process and $f(t, x)$ a $C^{1, 1}$ function. We define $$ Y_t = \int^t_0 f(s, X_s)\, \mathrm ds $$ Are there any general results pertaining to ...
3
votes
2answers
35 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
1
vote
1answer
19 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
0
votes
0answers
27 views

Real integral done by complex methods [duplicate]

$\int_{-\infty}^{\infty} \frac{cosx}{x^2+25} dx $ = $ \frac{\pi}{5e^5}$ Any ideas?
4
votes
1answer
23 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
1
vote
0answers
17 views

Bound on the mean value of function involving Hilbert transform

Consider the integral $$\int_{-\infty}^{\infty} x|A|^2_x\mathbb{H}(|A|^2_x) \ dx,$$ where $A=A(x,t)$ is a complex valued, compact function (I mean this in the heuristic sense that $A$ vanishes ...
0
votes
1answer
22 views

Volume of revolution on an area crossing the axis

Suppose a question asks for the volume of revolution about the x axis to be found on a piece of area enclosed between 2 graphs, where the area crosses the x-axis. In this case, the method involving ...
0
votes
0answers
22 views

Switching the order of integration [duplicate]

I need to switch the order of integration for the following function: $$ \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{x+y} f[x,y,z] dz dy dx $$ to the order: $$ \int_{?}^{?} \int_{?}^{?} \int_{?}^{?} ...
1
vote
1answer
55 views

How to solve this integration?

I want to solve this $$\int_0^w (b/x)^{a+1} e^{(cx-(b/x)^a)} dx$$ where $a$, $b$, and $c$ are arbitrary positive real numbers. Do i have to solve it numerically? I have no clue to solve this ...
0
votes
0answers
24 views

About a curve and its direction at every point

Consider the following class of problems: Consider a plane (or $\mathbb{R}^n$ in general). Let to any point of the plane corresponds a set of directions (set of unit vectors, I mean). Does there ...
1
vote
0answers
18 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
0
votes
0answers
16 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
0
votes
2answers
28 views

Which is the justification for this indefinite integral relation? [on hold]

Why is the following indefinite integral equation correct: $$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$ What are the necessary steps?
-2
votes
1answer
32 views

Area under the given curve [on hold]

The area under the curve $\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|}$ , $x$-axis and the ordinates at $x = -3$ and $x = 1$
0
votes
1answer
83 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
2
votes
0answers
53 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
3
votes
4answers
52 views

I'm stuck in this one of trig substitution for fuctions.

I got this: $$\int\frac{dx}{\sqrt{(4x^2-9)^3}}.$$ I know that the answer is: $$\frac{x}{9*\sqrt{4x^2-9}}+c.$$ And with the steps that I know about this type of substitution, I came up here, but.. ...
-5
votes
0answers
20 views

What method we use when n = odd for evaluate the integral using simpson's rule ??? [on hold]

hi any one can tell me What method we use when n = odd for evaluate the integral using simpson's rule ??? plz help....
6
votes
2answers
99 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
0
votes
1answer
37 views

Can you explain the result in this one, please?

I tried complete the square, but it doesn't work I got this: $\int\frac{xdx}{\sqrt{3-2x-x^2}}$ And I know that the answer is: $-{\sqrt{3-2x-x^2}}-\arcsin(\frac{x+1}{2})+c$
7
votes
2answers
68 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
0
votes
2answers
26 views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but… (see below)?

Is there a sequence $(f_n)$ of Riemann integrable functions such that $\lim f_n(x) = f(x)$ almost everywhere on $[a,b]$ and $\lim\int_a^bf_n$ does not exists in Riemann sense, but it does in Lebesgue ...
0
votes
1answer
9 views

Solving Poisson's Equation in 1-D for a point charge?

Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to get the familiar equation. This is what I did so far: So ultimately I am trying to solve this in 3 ...
4
votes
1answer
74 views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

For which values $\alpha \in \mathbb{R}$ is $f$ integrable? $$f: \mathbb{R}^2 \rightarrow \mathbb{R} : f(x, y) = x \frac{\ln(1 + x^2 + y^2)}{(x^2 + y^2)^\alpha} $$ if $ (x,y) \neq (0, 0) $ and $ ...
1
vote
2answers
37 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
1
vote
4answers
87 views

How does $\int (\cos(x))^{-2}dx$ equal to $\tan(x)$?

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
4
votes
5answers
96 views

How do I integrate$ \int\frac{1}{e^{2x}+e^x} \,dx $ [on hold]

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
2
votes
1answer
66 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
-1
votes
5answers
70 views

Integration of $\displaystyle \frac{e^x-1}{e^x+1}$ w.r.t. $x$ [on hold]

I tried a lot but unable to find out the solution of $$ \int\left(\frac{e^x-1}{e^x+1}\right)dx$$ Please solve it.
1
vote
0answers
21 views

Cauchy formula for repeated Lebesgue integration

Is there an equivalent of the Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for the following \begin{equation} f^{(-n)}(x) = \int_a^x ...
2
votes
1answer
33 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
1
vote
1answer
35 views

Evaluate the integral $\int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$

I am attempting to evaluate the integral (where $t \rightarrow \infty$) $$I(t) = \int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$$ which occurs in the calculation of the ...
1
vote
2answers
37 views

“Trig Substitutions”, I tried half- angle and trig indentity in this one, but doesn't work

I´m really lost in this one. $\int \sin^3 (2x) \cos^2 (2x) dx$ I know that the answer is: $\frac{1}{10}cos^5(2x)-\frac{1}{6}cos^3(2x) + c$ Please help
4
votes
3answers
97 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
1
vote
3answers
27 views

integration by parts of $25\, (1-\sin^{2}x)$

I need help solving this integration of parts problem. I've tried a few different solutions and keep getting the wrong answer. This question is in regards to this problem take the integral by parts ...
2
votes
1answer
22 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...
4
votes
1answer
43 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
3
votes
2answers
45 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
-1
votes
0answers
35 views

How to evaluate this definite integral? [on hold]

How to evaluate the integral $$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$ Here $a$ and $b$ are some positive real numbers smaller than $1$.
3
votes
2answers
89 views

Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important?

The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ ...
6
votes
3answers
218 views

Finding the definite integral of a trigonometric expression

Find the integral of $$ \int_0^{\frac{\pi}{2}}{{\sqrt{\sin(2\theta)}} \cdot \sin(\theta)d\theta}$$ I got $$I=\int_0^\frac{\pi}{4}{\sqrt{\sin(2\theta)} \cdot (\sin(\theta)+\cos(\theta))d\theta}$$ But, ...
0
votes
2answers
89 views

Can I solve this integral with a squared sum in it?

Title says it all. By now I have tried by hand and I think that it is indeed solvable, but I can't handle the very long terms. I tried to run the thing through SAGEs integrator: ...
2
votes
1answer
60 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
-1
votes
3answers
55 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
-8
votes
1answer
54 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!
0
votes
0answers
23 views

Double integral over a triangle

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function (derivable, integrable over all of $\mathbb{R}^2$). Let $T$ be a triangle in $\mathbb{R}^2$, defined by its vertices : $A=(x_a,y_a)$, ...
0
votes
0answers
21 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
votes
0answers
35 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
0
votes
3answers
38 views

fundamental theorem of calculus 2 [on hold]

Differentiate the following equation with respect to $x$: $$8 + \int_a^x \frac{f(t)}{t^2}\, dt = 2 x^{1/2}$$ Hence, find a function $f(x)$ and real number $a$ such that the above equation is true ...