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
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0answers
101 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
-4
votes
0answers
51 views

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$? [on hold]

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$?
1
vote
2answers
64 views

Volume of Solid Enclosed by an Equation

I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple ...
10
votes
1answer
515 views

The word “integral” in calculus unrelated to “integral” / “integer” in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
1
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2answers
46 views

How to evaluate the line integral $\int_C (y-z)\,dx+(z-x)\,dy+(x-y)\,dz$

How to evaluate the line integral $\int_C (y-z)\,dx(z-x)\,dy(x-y)\,dz$. The curve $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x-z=1$. I am really stuck on how to to do this ...
0
votes
1answer
35 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
1
vote
4answers
37 views

Find equation of curve

${dy \over dx}= (3x^2-a)^2$, where $a$ is a constant. Given that the curve has a stationary point at $(3,2)$, find the equation of the curve. I managed to get the equation $y=3x^3+3ax^2+xa^2$+c. I'm ...
0
votes
2answers
89 views

Proving that a function is Riemann Integrable

The usual definition to the Riemann integral is: for every $ε>0$, there exists $\delta$ such that if $P$ is a partition of $[a,b]$, and $\|P\|<\delta$, then $|S(f;P)-s|<\epsilon$. Then $f$ is ...
0
votes
1answer
48 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
2
votes
3answers
43 views

Find expression in terms of x

Knowing that $$\frac{dy}{dx}= k\cdot x^{\frac{1}{3}}$$ and given that it passes through points $(1,4)$ and $(8,16)$, find an expression for the path in terms of $x$. I found out that $$y= \frac34 k ...
0
votes
1answer
34 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...
0
votes
1answer
53 views

Reasons for different answers when finding area using Simpsons rule and numerical integration?

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
3
votes
0answers
41 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
0
votes
2answers
86 views

Evaluating $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates

I'm having issues solving $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates I made the ffg substitutions: $x=r\sin\theta\sin\phi, y=r\sin\theta \cos\phi, z=r\cos\theta$ Thus ...
2
votes
4answers
79 views

Solve $\int\frac{8x+9}{(2x+1)^3}\,dx$.

Do I split $\displaystyle\int\frac{8x+9}{(2x+1)^3}\,dx$ into partial fractions? Or do I use $(2x+1)^{-3}$ by itself? Not sure what to do. Please advice. The answer given is ...
0
votes
2answers
46 views

How is $ \frac{\sqrt{a}}{a+1} (0^{a+1}+1^{a+1}) $ equal to $ \frac{\sqrt{a}}{a+1} (-1)^a $

I am trying to integrate this equation $$ y = \int_{-1}^0 \sqrt{a} x^{a} $$ $$ y = \sqrt{a} \int_{-1}^0 x^{a} $$ $$ y = \frac{\sqrt{a}}{a+1} \int_{-1}^0 x^{a+1} $$ $$ y = \frac{\sqrt{a}}{a+1} ...
-2
votes
1answer
27 views

Find value of define integral with [closed]

Hi i need help for this problem, i very appreciate your sugerences. $$F(x)\text{=}\int ^{g(x)}_{0}\frac{dt}{\sqrt{1+t^{2}} } $$ And $$g(x)\text{=}\int ^{\cos x}_{0}[1+\sin t^{2}]dt$$ For $F'(π/2)$.
5
votes
2answers
363 views

demostration of interger part integration.

I need help for solving this demostration, I appreciate your suggestions very much. $$\begin{array}{rclr} \int ^{n}_{0}[x] dx= \frac{n(n-1)}{2} \end{array}$$ Pd. If you have any suggestion of a ...
0
votes
1answer
53 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
2
votes
2answers
32 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
8
votes
1answer
91 views

Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

I'm just wondering if the following integration is valid. \begin{array}{l} \int {\frac{1}{{\sqrt {{a^2} + {x^2}} }}} dx\\ {\rm{Let }}{u^2} = {a^2} + {x^2}\\ 2udu = 2xdx\\ \frac{{du}}{x} = ...
0
votes
0answers
20 views

How do you find the volume of a function rotated about the x axis along it's derivative? [closed]

So when you rotate a function, it is usually vertically. How do you rotate it around it's derivative, assuming that volume/area can overlap?
4
votes
4answers
93 views

Am I getting the right answer for the integral $I_n= \int_0^1 \frac{x^n}{\sqrt {x^3+1}}\, dx$?

Let $I_n= \int_0^1 \dfrac{x^n}{\sqrt {x^3+1}}\, dx$. Show that $(2n-1)I_n+2(n-2)I_{n-3}=2 \sqrt 2$ for all $n \ge 3$. Then compute $I_8$. I get an answer for $I_8={{2 \sqrt 2} \over 135}(25-16 ...
-4
votes
2answers
29 views

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $H(x) := \int_x^b f \ \ ,x\in I.$ [closed]

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $$H(x) := \int_x^b f, \ \ x\in I.$$ To find $H'(x)$ for $x \in I.$ I am stuck with the problem please help.
1
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0answers
34 views

Integral of a function which is everywhere discontinuous?

Yesterday, I tried to carry out a little thought experiment when it came to taking limits and have found that it has pushed my understanding of them to the breaking point. I tried considering the ...
3
votes
2answers
77 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
-2
votes
0answers
23 views

Integral Proof: Integral between h and 0, (h - z)(z - l)dz [closed]

How does the following give the result of l = h/3: Integral between h and 0, (h - z)(z - l)dz
1
vote
1answer
43 views

Double Integration word problem

In a certain metropolitan area, the population is approximated by the function: $$P(x,t)=\frac{\ 7274e^{0.5t}}{1+x}$$ Where $x$ is the number of miles from the center of the city, and $t$ is the ...
1
vote
0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
1
vote
1answer
42 views

Evaluate Double Integration

Evaluate $\iint−3x^2 dA$ over the region in the first quadrant bounded by the hyperbola xy=16 and the lines $y=x$, $y=0$, and $x=8$. I have drawn a picture, but I am still a little unsure on what to ...
2
votes
1answer
112 views

How can I prove the integral $ \int_{1}^{x} \frac{1}{t} \, dt $ is $\ln x $ with this approach?

I have been trying to find a proof for the integral of $ \int_1^x \dfrac{1}{t} \,dt $ being equal to $ \ln \left|x \right| $ from an approach similar to that of the squeeze theorem. Is it possible to ...
1
vote
2answers
29 views

function such that the sum of previous f(x) is smaller than f(x)

Just out of curiosity: is there a function $f$, such that $ \forall x, \sum_{x'<x} f(x') < f(x) $ sum or integral...
-1
votes
0answers
28 views

Calculus 2 - Rotating a region about an axis

I am having issues with the disk method and shell method when rotating a region around an axis. Example: The region bounded by $y=x^{\frac{1}{3}}, x=4y$, axis $x=3$ I am thinking that shell method ...
0
votes
2answers
38 views

Product Integral: Integrability

Given measure spaces $X$ and $Y$. Then it holds: $$\int_Y\int_X|\eta(x,y)|\mathrm{d}\mu(x)\mathrm{d}\nu(y)<\infty\implies\int_X|\eta(x,y)|\mathrm{d}\mu(x)<\infty\quad(y\in Y)$$ Can this ...
-3
votes
1answer
41 views

how to solve this problem by easy way [closed]

Can the following double integral: $$\iint xy(x+y+25)^{3/2} dx dy$$ be solved in an easy way?
-1
votes
2answers
37 views

Integration problem with $e$ and $\ln$

Can anyone help me solve this? I tried with integration with parts, but without luck. The function to be integrated is $$\frac{e^{x+\ln x}}{x}$$
0
votes
3answers
104 views

A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
2
votes
5answers
98 views

Mistake in evaluating $\int\dfrac{dx}{\ln(x)}$

Evaluate: $$I=\int\dfrac{dx}{\ln(x)}$$ My attempt: $$$$ $$I=\int \dfrac{x'}{\ln(x)} dx$$Integrating by Parts,$$\dfrac{x}{\ln(x)}-\int\dfrac{x}{(\ln(x))'}dx$$$$=\dfrac{x}{\ln(x)}-\int ...
1
vote
0answers
33 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
-3
votes
2answers
37 views

Crazy and difficult Limits and integration

This limit take from me much time to solve and finally I can't. So please help me to solve.. Find $L$ $ L =\displaystyle\lim_{x\to \infty} \frac{\displaystyle\int_{1}^{x} t^{t-1} ( t + tln (t) +1 ) ...
0
votes
1answer
71 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
2
votes
1answer
31 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
0
votes
0answers
21 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
6
votes
1answer
43 views

$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I$ [duplicate]

Let $I:=[a,b]$ and let $g: I \to \Bbb R$ be continuous on $I$. Suppose that there exists $K > 0$ such that $$|g(x)| \leq K \int_a^x|g| \ \ \forall x \in I.$$ Then $g(x) = 0\ \ \forall x \in I $. ...
2
votes
5answers
77 views

if we have $(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$ then $f(x) =x \ \forall x\geq0$.

Let $f: [0, \infty) \to \Bbb R$ be continuous and $f(x) \neq 0 \forall x>0$. If we have $$(f(x))^2 = 2 \int_0^xf, \ \forall x>0,$$ then $f(x) =x \ \forall x\geq0$. We have $(f(x))^2 = 2 ...
0
votes
1answer
11 views

$F(x) := (n- 1)x-\frac{ (n- 1)n}{2}$ for $x \in [n- 1, n), n \in \Bbb N$ using this result to evaluate $\int_a^b[x]dx.$

Let $F(x)$ be defined for $x \geq 0$ by $F(x) := (n- 1)x- (n- 1)n/2$ for $x \in [n- 1, n), n \in \Bbb N$. Show that $F$ is continuous and evaluate $F'(x)$ at points where this derivative exists and ...
6
votes
0answers
24 views

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
1
vote
0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
3
votes
3answers
50 views

How to find: $\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$?

How to find: $$\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$$ I tried multiplying it all out but I just ended up in a real mess and I'm wondering if there is something I'm ...
2
votes
1answer
60 views

Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a Riemann Integrable function. Let $\epsilon>0$. Show that there exist continuous functions $g,h:[0,1]\rightarrow \mathbb{R}$ such that $g(x)\leq f(x)\leq ...