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
18 views

Finding volume of a revolution

I want to find the volume of the revolution that occurs when the region bounded by $y = x^2$ and $y = 1$ is revolved around the line $y=2$. The problem is that it is not solid and I cannot understand ...
-1
votes
1answer
23 views

Find the centroid of the region under the graph of the function $ w(x) = 4.5 + a x^{3} $ between $ x = 0 $ and $ x = 5 $. [on hold]

I need to find the centroid to determine where the equivalent force is acting on the region under the graph of $ w $ between $ x = 0 $ and $ x = 5 $. The given information is $$ w(0) = 4.5 ~ ...
4
votes
2answers
76 views

Finding $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$

I wanted to find $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$, so I substituted $t=\sin\theta$ and got $\int\frac{\cos^2\theta}{1+\sin^2\theta}d\theta$; but I'm not sure what the best way to proceed from here ...
2
votes
0answers
12 views

double integration with the same variable

I have the integral that I want to resolve. To calculate the flux of the electric machine, I have the following formula: $v_s= R_s \cdot i_s + \frac{\Phi _s}{dt}$ where $v_s, i_s, \Phi _s$ are ...
0
votes
1answer
47 views

How to find the integral with $\sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } $ in the denominator?

How to evaluate $$\int { \frac { 1 }{ \sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } } } +\frac { \log { (1+\sqrt [ 6 ]{ x } ) } }{ \sqrt [ 3 ]{ x } +\sqrt { x } } dx$$ I'm not being able to make the right ...
0
votes
3answers
25 views

other form of uv notation of integration by parts

So integration by parts looks like this $\int u\, dv = uv - \int v\, du$ But I have often seen it like this: $\int uv\,dx = u \int v\, dx - \int (u'\int v dx )\, dx$ I cannot prove this. $ uv ...
4
votes
3answers
121 views

What do these symbol mean?

I always see these symbols and others like it when looking at really advanced maths. I have yet to learn anything about it. I was wondering if someone could explain briefly what they are used for. ...
4
votes
2answers
43 views

Integrate with $-d(x/y)$

Here's an integral which I encountered that uses some unfamiliar notation for me: $$\int-\frac{d(x/y)}{\sqrt{1+(x/y)^2}}$$ What does this mean? I don't have much of an idea. Edit: This problem is ...
1
vote
1answer
19 views

Finding the equation of a curve where the gradient is $ax + b$ at all points.

The gradient of a curve is $ax + b$ at all points, where $a$ and $b$ are constants. Find the equation of the curve given that it passes through the points $(0,4)$ and $(1,3)$ and that the tangent at ...
5
votes
4answers
99 views

Finding $\int\frac{1}{x^{11}+4x^6}dx$

I wanted to find out if there is an easy way to evaluate $\displaystyle\int\frac{1}{x^{11}+4x^6}dx$. I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find ...
-2
votes
0answers
13 views

What would be line integral along path number (iv) [on hold]

In the above image what should be the line integration along path iv. Thanks.
2
votes
0answers
35 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
-2
votes
0answers
18 views

Surface Integral of cone [on hold]

How would i calculate the surface integral of this the radius of the cone being 28.25 Thank you
9
votes
1answer
86 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
1
vote
1answer
38 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
2
votes
5answers
353 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
0
votes
0answers
46 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
1
vote
1answer
43 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
1
vote
1answer
47 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
2
votes
2answers
60 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
1
vote
1answer
34 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...
3
votes
2answers
87 views

How to evaluate $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$?

Can anyone suggest the method of computing $\int_0^1 \ln(\frac{1+x}{1-x}) \frac{dx}{x} = \frac{\pi^2}{4}$ ? My trial is following first set $t =\frac{1-x}{1+x}$ which gives $x=\frac{1-t}{t+1}$ ...
2
votes
4answers
62 views

finding $\int {(2x + 5)^2}$

After slowly getting the hang of differentiation I have moved onto integration and I can't seem to understand this one. I know the answer is $$\frac{4x^3}{3} + 10x + 25x + C$$ I understand that ...
1
vote
3answers
48 views

Change the order of expectation

Sorry this might be a silly question, but I'm kind of confused and really want to make sure I'm correct. Let $v_1,v_2,\dots,v_n$ be $n$ i.i.d. random variables with the same range of ...
1
vote
0answers
42 views

Integral tending to an integral for $\pi$

I am examining: $$\int_0^1 (1-ax)^{1/2} dx$$ If we differentiate: $$\dfrac{d}{dx} \left[\dfrac{-2(1-ax)^{3/2}}{3a}\right]$$ we get to the function in the integral. The idea now is consider various ...
1
vote
0answers
9 views

Quadruple integral of the solution to a new type of fractional differential equation

Let $\text{D}$ denote the differential operator, and $\text{D}^n$ the $n$th application of $\text{D}$ (i.e. the $n$th derivative) for any positive integer $n$. Note that $\text{D}^0 = ...
-1
votes
3answers
46 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
9
votes
0answers
86 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
-7
votes
1answer
44 views

Solve the integral $\int _0^1\:\frac{\sqrt[3]{x}+1}{1+x}dx$ [on hold]

Solve the integral $$\int _0^1\:\frac{\sqrt[3]{x}+1}{1+x}dx$$
1
vote
1answer
30 views

Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
1
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0answers
24 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
-2
votes
2answers
69 views

Unable to understand integration formula $\int 2x dx=x^2+2$ [on hold]

$$\int 2x dx=x^2+2$$ How? In all books this step is done directly. I am a beginner in integration. Please explain with all the necessary steps.
0
votes
0answers
27 views

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (t-s)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
1
vote
1answer
18 views

integrability of discontinuous functions

The FTOC states that if $f$ is continuous on $[a,b]$ then it is integrable. If $f$ is not defined at certain points of $[a,b]$ we can often give meaning to an improper integral. But under what ...
0
votes
0answers
26 views

Cauchy's Integral Formula Question- Calculating an integral with z^4 + 16 on the denominator

I think the first part of this question is okay. For the second part, I have found the roots and then calculated the absolute difference between these roots and i and, as they are all greater than ...
0
votes
3answers
24 views

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $ for some $ A \in \mathbb{R} $ Is it sufficient to say that if the derivative of a function contains itself, then it must be the exponential ...
0
votes
0answers
37 views

MAple 17 won't evaluate my integral

I type this into maple and it won't evaluate it: $$ \int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy $$ I've also tried $$ evalf(\int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy)$$ It just returns for both cases $$ ...
0
votes
1answer
22 views

Balancing the area of curves - integration

A linear equation needs to be found $y=ax+b$ with a slope of a maximum value of $10$ degrees that will balance out the area above the line and below the line for the function ...
0
votes
1answer
23 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
1
vote
2answers
53 views

How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

$\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$ First i try $\ln(x)=t$ so that $\frac{1}{x} dx =dt$ then integral becomes \begin{align} &\int_{-\infty}^{0}\frac{e^t-1}{t} (e^t dt) = - ...
6
votes
3answers
163 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First i try the subsititue $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
-3
votes
2answers
38 views

Integral of $f(x)= x\sin(n \pi \cdot \frac{x}{L})$ [on hold]

How do I go about integrating $x\sin\left(n \pi \cdot \frac{x}{L}\right)$ with respect to $x$? (which methods etc?) Thanks in advance for tips/advice/solutions.
0
votes
2answers
43 views

Definite integral from -1 to 0 [on hold]

How would I evaluate this definite integral $$ \int_{-1}^{0}\tan x dx- \int_{-1}^{0}\sin^2 x dx $$ All i need to know is what to do when an integral is on an interval of -1 to 0. I could do this ...
3
votes
0answers
47 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
0
votes
0answers
19 views

Using the general slicing method to find the volume of a semi-circle whose cross sections are squares.

In finding the volume of a solid, described below, I was close in finding the equation, but neglected a coefficient. Please see the question below. Use the general slicing method to find the volume ...
1
vote
1answer
33 views

How to integrate hydrostatic force on a two dimensional shape?

I'm so confused this question is very different from the other hydrostatic force questions and I think I am misunderstanding the question. I am primarily concerned with 15 because I somehow managed ...
3
votes
5answers
564 views

Don't understand the Fundamental Theorem of Calculus

If $f$ is continuous on $[a, b]$ and defining $$ F(x) = \int_a^x \! f(t) \, dt $$ for $x \in [a, b]$, then $F'(x) = f(x)$ for $x \in (a,b)$. I don't understand what function the variable ...
0
votes
0answers
31 views

how to use complex integration to calculate $\int_0^{\pi}(1/a+\cos(x))dx$?

I have so far replaced $dx$ by $1/zi \ dz$, but I don't know how to deal with $\cos(x)$
1
vote
1answer
32 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
-1
votes
0answers
24 views