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 ...

learn more… | top users | synonyms (3)

1
vote
1answer
27 views

Estimation of Integral $E(x)$

How can we prove $$\frac{1}{2}e^{-x}\ln\left(1+\frac{2}{x}\right)<\int_{x}^{\infty}\frac{e^{-t}}{t}dx<e^{-x}\ln\left(1+\frac{1}{x}\right)\;, x>0$$ $\bf{My\; Try::}$ Let $\displaystyle ...
0
votes
0answers
55 views

Trivial equation but with integral

Let's consider the following equation: $$\int_{a}^{x} f(t)dt = K$$ where $a, K \in \mathbb{R}$. Suppose that $a, K$ and the function $f$ are known and that the equation should be used to determine ...
0
votes
1answer
22 views

Numerical integration in Matlab (Gaussian 3 point quadrature)

Write a Matlab function that applies the Gauss three point rule to N sub-intervals of $[a, b].$ The input parameters should be the name of the function being integrated, $a, b,$ and $N$. Attempt: ...
3
votes
1answer
86 views

Looking for a closed form of $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}} \, dx$

I am looking for a closed form of this integral for reals $a,b>0$ and integers $n,m>0$ $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}}$ I read about $I(2,2)$ in the book Irresistible ...
0
votes
0answers
6 views

Can quaternions be used to represent rotation rate?

A quaternion is a useful tool for representing a rotation, or change in attitude. If a quaternion $q$ represents a rotation, and $v$ a vector, then $v'=qvq^*$ rotates the vector, where the multiply ...
8
votes
2answers
70 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
9
votes
5answers
481 views

538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
18
votes
4answers
668 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
1
vote
1answer
28 views

Kernel and Image of an integral.

Im struggling to answer a question where $F: P_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R}) $ $$F(f)(x)=\int^{x+1}_{2-x} (1-t)f(t) dt$$ So to find the Kernel do i set the integral equal to 0 and ...
0
votes
1answer
22 views

Problem solving a partial derivative with a integral.

Good night, i have a serious problem solving this partial derivative: $f(x,y)=\int_{y}^{x}e^{t^{2}}dt$ I don't know how i can start this, please give me a help, don't do it the exercise, only explain ...
0
votes
3answers
25 views

Surface integral of a scalar over a unit cube.

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$ where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$ Honestly, I don't know what to do. All I know is that you have to ...
11
votes
4answers
656 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
1
vote
1answer
48 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where ...
0
votes
0answers
24 views

Show $R(x)=o(x^3)$

I got $$R(x)=n! \, x^n \int _0^{\infty} \frac{(-1)^n}{(1+xt)^{n+1}}e^{-t} \, \, dt$$ is this correct? I am stuck on the second part. Am I right in saying the first condition is ...
0
votes
2answers
58 views

help for an integral

I need help calculating this integral: $$\int_0^x \frac{2(e^{\gamma u}-1)}{(\gamma+\kappa)(e^{\gamma u}-1)+2\gamma} du$$ I tried with the integration by parts but the situation seems to get ...
2
votes
1answer
30 views

Magnetic field by current in an infinite cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder of radius $R$, with its axis coinciding with the $z$ axis, entirely enclosed by the cylinder's lateral surface. Then, for any constant ...
-2
votes
0answers
31 views

Integration possible or not. [on hold]

What are the conditions that a function must satisfy such that it can be integrated?
1
vote
0answers
27 views

unbounded solution, lim inf of trace,

Show that if $\lim \inf_{t\rightarrow \infty} \int_{t_0}^t \operatorname{tr}\left(A(s)\right)ds= \infty $ then the linear first-order system $x'(t)=A(t)x(t)$ where $A \in C\left(I, \mathbb{R}^{n\times ...
-3
votes
1answer
47 views

Double integral involving $y /\sin y$ [on hold]

Let $f\colon \Bbb R^2\to \Bbb R$ be defined by $${f(x,y)= \begin{cases} \frac y{\sin y}, &y\neq 0\\ 1, &y=0 \end{cases}}$$ Then the integral $${\frac ...
-2
votes
3answers
87 views

find the value $\int_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2dx$ [on hold]

Find the $$I=\int_{0}^{+\infty}\left(\dfrac{x^2}{e^x-1}\right)^2dx$$ Let $x=\ln{t}$,$$I=\int_{1}^{+\infty}\dfrac{\ln^4{t}}{t(t-1)^2}dt$$
4
votes
0answers
43 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq ...
1
vote
1answer
756 views

Centroids and volume

I'm not quite sure what i'm doing wrong here.. Find the centroid (bar-x,bar-y) of the plane region defined by Then use Pappu's theorem to find the volume V of the solid obtained by rotating the ...
3
votes
2answers
44 views

Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$. $x$ and $y$ range from $0$ to $2$. $$\int_0^2 \int_0^2 y-3 \,dy\,dx $$ is correct?
0
votes
1answer
20 views

the angle of a shifted cone and its equation

How to get the angle of the cone if it is shifted ?? $$z= 1-\sqrt{ x^2+y^2}$$ $$\rho \cos(\phi)=1- \rho \sin(\phi)$$ then I can not get the angle ! Do I have to get the angle when it is not shifted ...
0
votes
1answer
34 views

Double Integral Range

we want the Surface between the two equation: $r= 1 + \cos\theta$ $r = 1$ (circle) we can use a duble integral to solve this: $$ S = \int_{-\pi/2}^{\pi/2} \int_1^{1+\cos\theta} r \,dr\,d\theta ...
-3
votes
0answers
38 views

Please help me with these 4 questions, thanks. [on hold]

This is the image with the questions: (Large version)
0
votes
1answer
18 views

“Nonlinear cosine” integral

Let $\alpha > 1$, $\xi \in\mathbb{R}$. and $\chi_A$ be the characteristic function of the set $A$. Are there some known ways of computing (or estimating in terms of $\xi$) of this kind of ...
1
vote
4answers
54 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral ...
0
votes
2answers
42 views

Integration substituion

I need to use a suitable substitution to show $$\int {dx \over \sqrt{a^2+x^2}} = \text{arcsinh} \left(\frac{x}{a}\right)+C$$ but I am not sure what substitution to use. Any help would be great. ...
0
votes
3answers
77 views

How to check if the integral of a function is correct with a calculator?

As a calculus student I would like to know if there is any way to verify the "correctness" of an integral of a function using a calculator. I mean if have something like f(x) = cos(2X), the integral ...
0
votes
0answers
14 views

Integral of implicit function - geometric meaning

What is the geometric meaning of implicit function $$f(x,y) = 0$$ integral? Is it the same as for explicit function, eg. area, volume etc.. and we are computing it for the $0$ result, or is there ...
1
vote
2answers
44 views

Definite integral of a positive continuous function equals zero?

Let's calculate $$\int_0^{\frac\pi 2} \frac {dx}{\sin^6x + \cos^6x}$$ We have $$\int \frac {dx}{\sin^6x + \cos^6x} = \int \frac {dx}{1 - \frac 34 \sin^2{2x}}$$ now we substitute $u = \tan 2x$, and get ...
0
votes
1answer
42 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
1
vote
2answers
35 views

Am I interpreting antiderivatives the right way?

Here the antiderivative of $f(x)=x^2$ is $F(x) = x^3/3 + C$. If the constant of integration is $C = 0$, then for any $x$, $F(x)$ would give me the area under the curve of $f(x)$ from $0$ till $x$ ...
0
votes
1answer
22 views

How can I understand $x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds$?

I am trying to understand this integral form of the ordinary differential equation: $$x(b)=x(a)+\int_a^{b}f(s,x(s))\,ds\quad\text{for }a\leq t\leq b$$ I tried to pick a concrete example: Let ...
2
votes
1answer
43 views

TIFR GS 2015 computer science: $G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$

Following expression was asked to be evaluated in TIFR GS 2015 exam, $$G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$$ where $x \in [0, 1]$ and $f(x)$ be any real valued continuous function. ...
2
votes
2answers
47 views

Estimation for integral

I am trying to show that $$\lim_{R\to\infty}\left | \int_{0}^{R}e^{-(R+iu)}(R+iu)^{s-1}\,\mathrm{d}u \right |=0$$ and $$\lim_{R\to\infty}\left | \int_{0}^{R}e^{-(u+iR)}(u+iR)^{s-1}\,\mathrm{d}u \right ...
3
votes
2answers
36 views

Evaluating $ \dfrac{1}{2 \pi} \int_{-\infty}^\infty \int_{-\infty}^\infty e^{tuv} e^{-u^2/2} e^{-v^2/2}\ du\ dv $

Solving a probability problem I came across this integral: $$ \dfrac{1}{2 \pi} \int_{-\infty}^\infty \int_{-\infty}^\infty e^{tuv} e^{-u^2/2} e^{-v^2/2}\ du\ dv $$ Can you explain how to ...
0
votes
2answers
54 views

To find the initial value problem $y(x) = 1 + \int_{0}^x (t-x) y(t)\,dt$ [on hold]

The initial value problem corresponding to the integral equation $$y(x) = 1 + \int_{0}^x (t-x) y(t)\,dt$$ is?
0
votes
3answers
30 views

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$

Examine convergence of $\int_0^{\infty} \frac{1}{x^a \cdot |\sin(x)| ^b}dx$ for $a, b > 0$. There are 2 problems. $|\sin(x)|^b = 0$ for $x = k \pi$ and $x^a = 0$ for $x = 0$. We can write ...
0
votes
0answers
29 views

Double Integration problem?

$$M = \int_0^1[\int_0^x(x+y^2)dy]dx$$ I need to find the function describing the surface mass density? I thought this would just be $$x+y^2$$ Im not sure if im right. Then I need to change the ...
2
votes
4answers
52 views

Use the definition of $\ln(x)$ as an integral to show that $f(x)=\frac{ln(x)}{x^2} \leq 1/x$ for all $x\geq 1$.

As the title says, if we let $$f(x)=\frac{ln(x)}{x^2}$$ I know that $$ln(x)=\int_1^x \frac{dt}{t} dt$$ Since $x^2 >0$ we can rewrite the question as $ln(x) \lt x$ $\forall$ $x\ge1$ How do we ...
0
votes
2answers
24 views

How to determine the convergence radius and intervale of convergence from this sum

I have to find the converge radius and interval of convergence for the serie, I've tried the hHadamard criteria but I had no succes. I hope you can help me. $\sum_{n=1}^{\infty}(2+(-1)^n)(1+x)^{n-1}$ ...
-3
votes
2answers
46 views

Find the Taylor's Series for $f(x)=x^3-10x^2+6$ about $x_0=3$ [on hold]

Please help me. I want a solution for this question Find the Taylor's Series for $$f(x)=x^3-10x^2+6$$ about $x_0=3$.
16
votes
4answers
1k views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
2
votes
1answer
23 views

Volume bounded by sphere and cone

Is the volume of solid bounded below by the sphere $$\rho=2 \cos(\phi)$$ and above by the cone $$z=\sqrt{x^2+y^2}$$ will be gotten by the following integration ? in cylindrical form : $$V ...
0
votes
1answer
346 views

Tetrahedron volume in the first octant

The surface is given: $xyz = 2$ It is in the first octant so $x > 0, y > 0, z > 0$. The tangent plane taken at any point of this surface binds with the coordinate axes to form a ...
0
votes
0answers
44 views

Double Integration?

I'm having trouble integrating this function. $$\int_0^{\frac{\pi}{4}} \int_{0}^{1} \frac{y\ln(1-x^2-y^2)}{x-\sqrt{x^2+y^2}} \,dy\, dx$$ I just need to double integrate over the domain but when I ...
1
vote
2answers
29 views

Double integration help?

I have a question where I need to double integrate over the domain D. The domain D is bounded in the first quadrant by the following lines: $$x=0$$ $$y=x$$ $$x^2+y^2=1/4$$ What would the domain of ...
1
vote
2answers
60 views

Integration by substitution limits confusion

If I have the integral: $\displaystyle\int_{0}^{\infty}t^{-\frac{1}{2}}e^{-t} dt$ Am I allowed to make the substitution $t=x^2$, because I am then not sure what the limits of integration would be as ...