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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56 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
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1answer
43 views

Determine the volume of $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$

Let $f\in L^2(\mathbb R)$ and $f\geq0$. Determine $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$. "Normal" substitution $(x=rcos(\phi),y=rsin(\phi))$ did not help a lot, since I dont have ...
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2answers
162 views

Area enclosed by cardioid using Green's theorem

Let $$\gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix}, \qquad t \in [0,2\pi].$$ Find the area enclosed by $\gamma$ using Green's theorem. So the area enclosed by ...
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1answer
44 views

Trigonometric integral (arctg)

I have a problem with this integral. $$\int \text{arctan}(x-2)dx =\text{ }?$$ I tried integration by parts but it doesn't lead to right result.
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5answers
50 views

How to caluclate the integral of $\int \frac{1}{\sqrt{4x^{2}+1}}dx$ using a trig substitution?

I am trying to determine the following integral: $\int \frac{1}{\sqrt{4x^{2}+1}} dx$ using a suitable substitution. My progress: let $x = \frac{1}{2} \tan \theta$ $dx = \frac{1}{2}\sec^{2} \theta ...
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0answers
31 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
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2answers
93 views

How to solve such an integration analytically?

$\displaystyle\int^{2\pi}_{0} e^{ia \cos{\theta}}d\theta$ where $a$ is some constant. Can it be solved with some substitution? I tried it by expanding the exponential series but that was not proper ...
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1answer
25 views

Volumes of revolution?

The point $P(a,b)$ lies on the curve $y=\mathrm{arsinh}\,x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated $2\pi$ radians about the $x$-axes the ...
4
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2answers
88 views

Why does $\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$?

I was wondering why the following is true: $$\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$$ It is easy to obtain this result by doing a trig substitution but it's messy and not ...
2
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2answers
101 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
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2answers
44 views

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$ I've seen a proof which goes like this. $$ \lim_{x\to\infty} \frac{\frac{\arctan x}{x^2}}{\frac{1}{x^2}} = \frac{\pi}{2} > ...
3
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1answer
70 views

finding $\int\frac{1}{(t^2+25)^2} dt$ without trig substitution

Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate $$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$ without using ...
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2answers
51 views

Integration of the following

What is the definite integral of $$ \int_0^1 \left(\frac{g(x)}{f(x)}\right)'\cdot\frac{1}{g(x)}\,dx, $$ where the conditions are as follows: $f(0) = 2 $ $f(1) = 3 $ $f'(x) $ is continuous For all ...
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1answer
59 views

Evaluate the integral $\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$

Evaluate the integral $$\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$$ My Try: substituting $t = 2^x$ we get: $$\ln 2 \int_2^\infty \left(\frac{1}{2}\right)^t dt = \frac{\ln 2}{\ln 0.5} \left( ...
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0answers
55 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
2
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2answers
316 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
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2answers
79 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
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1answer
50 views

Why doesn't this method for integration by parts work?

So here is what I did first. $$∫16\ln(x^{1/3})dx$$ move the constant $16$ out $$16∫\ln(x^{1/3})dx$$ use properties of logarithms to rewrite natural log of cube root of $x$ as $\ln x$ divided by $3$ ...
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0answers
15 views

Transformation of the infinitessimal integration variable under a coordinate transformation

I always get confused when I'm facing the 3D integral over space and have to do a coordinate transformation on the given function to solve the integral. Do some of you have tips/trick how to ...
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1answer
38 views

Find all differentiable equations using Cauchy-Riemann equations

Let $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. I want to use the Cauchy Riemann equations to find all differentiable functions of the form $$Re( h(z))=2x^2+2x+1-2y^2$$ So I used the C-R equations with ...
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1answer
95 views

Fundamental theorem of calculus, differentiable at the endpoints.

One version states: Let f be a continuous real-valued function defined on a closed interval $[a,b]$. Let f be the function defined for all x in $[a,b]$, by $F(x)=\int_{a}^xf(t)dt$. Then, F is ...
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0answers
47 views

Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
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2answers
43 views

How would I start to solve this?

I need to calculate the derivative of $F(x)=\int_{f(x)}^{f^2(x)}f^3(t)dt$. Usually for a derivative of an integral I would plug the upper bound and lower bounds into $f(t)$ then multiply each by their ...
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1answer
42 views

Evaluate $\int^\infty_0 t^{a+b-1}(t+1)^{-b-1} U(a+2,a-b+2,ct)dt$

Evaluate $$ \int^\infty_0 t^{a+b-1}\left(t+1\right)^{-b-1} U\left(a+2,a-b+2,ct\right)dt $$ under the condition $a>0$, $b>0$ and $c>0$, where $U(\cdot,\cdot,\cdot)$ denotes the ...
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0answers
33 views

Find the Volume of the Solid--Cylindrical Shells

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $$ y = 2x^2, x = 1, y = 0 $$ about the x-axis I can't seem to get this. It's in a ...
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1answer
23 views

ODE: Why do we change our variable here?

I was trying to solve a matrix equation $\dot x = Ax + Bu$ Rearranging yields $\dot x - Ax = Bu$ Let $I = e^{-At}$ our integrating factor so $d(xe^{-At})/dt = e^{-At}Bu$ Then $xe^{-At}$ = $x_0 ...
3
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3answers
516 views

How can I prove this integral is equal to f(0)?

Given that $f$ continuous over $[-1,1]$, how can I show $\lim_{x \to 0}\frac{1}{x}\int_0^xf(t)dt = f(0)$? I know the limit of $\frac{1}{X}$ doesn't exist at 0, and it's negative infinity from the ...
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2answers
81 views

Integrate $2^{x^2}$

Can someone please show me the integral $\int 2^{x^2}dx$? I know that the integral of a constant $b^x$ would result to $$\frac{b^x}{\ln b}$$, so would that mean that the function be $2^{x^x}$ and ...
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1answer
54 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
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1answer
137 views

Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$. I know this can be done by the sv-calc method of volumes of revolution but I ...
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1answer
43 views

Evaluate an integral quickly

Evaluate the integral $$\int \sqrt{x} \ln(1+x)dx $$ so we should start with the substitution: $t=\sqrt{x}$ $$ \int t\ln(1+t)dt2t = 2\int t^2\ln(1+t)dt $$ From here, it seems reasonable to ...
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0answers
19 views

Approximating ArcCos(x) without Radicals

Take $$f(x)=2x\arccos\left(\frac{x^2+d^2-1}{2xd}\right)$$ and try and find $$ I(x)=\int_{d-1}^{3}dx f(x) \sqrt{\left(\frac{x-1}{x}\right)}\left(3-x\right)^3 $$ You'll find the result is messy (see ...
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0answers
101 views

Find the volume bounded by the sphere $x^2+y^2+z^2=4$ and $x^2+y^2-2x=0$

This question appeared on my calculus exam yesterday. I don't know how to do it: Find the volume bounded by the sphere $x^2+y^2+z^2=4$ and $x^2+y^2-2x=0$. My attempt: First, I realised that the ...
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1answer
67 views

finding the parametric path for line integral

Calculate the work done by the force field $F(x,y,z)=(y^2,z^2,x^2)$ along the curve of intersection of the sphere $x^2+y^2+z^2=1$, the cylinder $x^2+y^2=x$, and the halfspace $z>0$. The path is ...
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1answer
46 views

Calculate $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$

I have to calculate (if it exists) $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$. I think I have to use Lebesgue dominated ...
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1answer
74 views

Simple Integral Involving Radicals: Why Does Mathematica Fail?

I have $$\int_{d-1}^{3}\textrm{d}x\left(3-x\right)^3 \sqrt{\left(\frac{2(x-1)}{x}\right) \left(x-\left(d-1\right)\right)}$$ but despite this looking like a simple integral involving fractional ...
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1answer
90 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
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1answer
29 views

Separation of variables and why integration of 1/x terms gives ln|x|

So assuming I got something like $$x'(u)=-\frac{x}{u}$$ which gives me then (with separation of variables) $$\int\frac{dx}{x}=\int{-\frac{du}{u}}$$ So my question is: Why do I get $$ln|x|=-ln(u)+c, c ...
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1answer
47 views

Show that integral of Gaussian distribution is 1

Under a normal distribution, μ = 0 and σ = 1, but when then integrating this equation, I get an error function. Without using Riemann sums, how can I prove that this equation = 1? I have only had ...
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3answers
47 views

Integral $ \int \frac{1}{x^{1+a} (1-x)^{1-a}} dx~,~a \gt 0$

The following integral is part of a large problem I'm trying to solve and I'm stuck. I'd appreciate some guidance. I would like to know how to compute integrals of the form $$ \int ...
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2answers
299 views

Find $\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
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2answers
76 views

How to evaluate the integral $\int x^2/\sqrt{4-x^2}\,dx$?

How to compute this integral? $$\int \frac{x^2}{\sqrt{4-x^2}}dx$$ If there were $x$ instead of $x^2$ in the numerator I know how to do a substitution $y=4-x^2$. But this doesn't help with the $x^2$.
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0answers
35 views

Why is this integral involving the mean value function zero?

Let $u$ and $v$ belong to $H^1(\Omega \times (0,\infty))$ on a bounded domain $\Omega$. Define $$(Au)(y) := \frac{1}{|\Omega|}\int_\Omega u(x,y)\;\mathrm{d}x.$$ We have that $Au \in H^1(0,\infty)$. ...
5
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2answers
125 views

Limit $I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$

Im a new participant in this mathematical forum, so this is one of that i couldn't solve it. $$I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$$ I've ...
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0answers
20 views

Approximating the Arc Length of a Regular Curve with a Broken Line

Question: Suppose $\alpha:[a,b]\to\mathbb{R}^3$ is a regular curve segment. Prove that, for every $\epsilon>0$, there exists $\delta>0$ such that, for any partition ...
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0answers
96 views

Importance Sampling of 2D constant piecewise function convertible to 1D?

So I have a constant piecewise 2D function (luminance values of pixels of an image) that I am writing an importance sampling algorithm for. I was going to write my algorithm by first sampling the 1D ...
2
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1answer
81 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
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1answer
49 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
2
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2answers
52 views

Prove that there exists $x_0\in [a,b]$ such that $ \sum_{i=1}^{n} k_i \int_{x_0}^{x_i} fdt=0$

Let $f$ is a continuous function on $[a,b]$, $x_1,x_2,\ldots,x_n\in [a,b]$, $k_1,k_2,\ldots,k_n>0$. Prove that there exists $x_0\in [a,b]$ such that $$k_1\displaystyle \int_{x_0}^{x_1} ...
1
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0answers
29 views

Integral of symmetric function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be such that $f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for every $n$-permutation $\sigma$, and suppose that ...