Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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2
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3answers
76 views

Show the Beta function $B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt$ is defined for $x,y > 0$ without actually integrating

An old exam question: Show that $$B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt$$ exists for all $x,y>0$. I'm sure because of time allotment that it's not in the scope of the question to actually ...
0
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1answer
31 views

Verification: using polar coordinates correctly?$\int_Kx^2(x^2+y^2)\mathop{d(x,y)}$

Doing some old exam questions, this popped up: Calculate $$\int_Kx^2(x^2+y^2)\mathop{d(x,y)}$$ where $$K = \{(x,y)\in \mathbb{R}^2\vert x^2+y^2\le 1\}$$ I'm sure polar coordinates are the way to go, ...
0
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0answers
28 views

Improper integral criterion

im trying to solve the following by the limit comparison theorem. Problem: $\int_{0}^{1} \frac{\sin(x)}{x^{3/2}(1-x)^{2/3}}dx$ is convergent or divergent? since its a type II with 2 indeterminations ...
2
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0answers
33 views

Can the Leibniz integral rule be used on integrals with singular endpoints?

I'd like to know if the Leibniz integral rule has an extension or generalization that can handle convergent integrals whose endpoints are singular. This post attempts to ask a similar question but ...
0
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2answers
74 views

Evaluate $\int \frac{1}{x\ln(n)}dx$ [closed]

I could have solved this by substitution, but the ‘n’ is confusing me. How should I proceed?
-3
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0answers
35 views

how can I calc the area of this triangle when I'm given this double integral? Tried Green's theorem

Basically we were given this integral:- $$\int (x^3dx-y\,dy)$$ and the points of the triangle are $(0,0) (1,1) (1,2)$ I tried using Green theorem but it will just give me $0$, so that can't be ...
5
votes
4answers
65 views

Getting different answers for an integral: $\frac{1}{2}x-\frac{3}{2}\ln{|x+2|}+C$ vs $\frac{1}{2}x-\frac{3}{2}\ln{|2x+4|}+C$

Problem: $$\int\frac{1}{2}-\frac{3}{2x+4}dx$$ Using two different methods I am getting two different answers and have trouble finding why. Method 1: $$\int\frac{1}{2}-\frac{3}{2x+4}dx$$ $$\int\...
0
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1answer
65 views

Volume intersecting spheres using triple integrals

Hi I have been looking for this triple integral almost a week now, I can not find a good integral at the end, I need to solve this with triple integrals, I know there are other ways to calculate it. ...
1
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1answer
20 views

Substitution methods of first order linear eqautions [problem help]

Okay so I am having some computational issues towards the end of the problems Problem 1: $y' = (4x+y)^2$ $z = 4x + y$ $y = z - 4x$ $y' = \frac{dz}{dx} -4$ $z^2 = \frac{dz}{dx}-4$ $z^2 + 4 = \...
1
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3answers
65 views

Show that $\int_0^b f(x)(b-x)\,dx=\int_0^b(\int_0^x f(t)\,dt)\,dx$

Let f: $\mathbb{R}\rightarrow\mathbb{R}$ be an arbitrary continuous function. Show that: $$\int_0^b f(x)(b-x)\,dx=\int_0^b(\int_0^x f(t)\,dt)\,dx$$ I have gotten relatively close using integration ...
1
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2answers
53 views

Is it possible to construct a 3D equivalent of Gabriel's Horn in a higher dimensional space?

Gabriel's Horn has the interesting property that it is an infinite surface area bound within a finite volume. I was wondering if there was an extension of this to 3D space in a higher dimensional ...
58
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8answers
4k views

Graphs for which a calculus student can reasonably compute the arclength

Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by $$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$ For many choices of $f$ this can be a ...
1
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0answers
38 views

Non-integrable bounded function which has antiderivatives

I would like to construct a bounded, real function $f:[a,b] \to \mathbb{R}$, which is not Riemann integrable, but has antiderivatives. I can easily construct an unbounded function with this property, ...
5
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3answers
109 views

Integrate ${\int\sqrt{1 + \sin\frac{x}2}\,\mathrm{d}x}$

So I was doing a integral question and I stumbled upon this question. $\displaystyle{\int\sqrt{1 + \sin\left(\frac x2\right)}\,dx}$ In order to solve it I did the following: I took $u = \frac12x$ ...
4
votes
1answer
44 views

Limits of integration for parametric equation

For the picture attached I am wondering why I cannot take the limits from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. If I take those limits, Sine vanishes (in the second last step) and the answer varies ...
0
votes
1answer
63 views

Integration of $\frac{1}{1-\cos(\alpha)\cos x }$

Integration of $\dfrac{1}{1-\cos(\alpha)\cos x }$ w.r.t $x$ How to do this problem? I was trying to reduce it $\dfrac{1}{1-\cos(\alpha)\cos (x) }=\dfrac{\sec (\alpha)}{\sec (\alpha)-\cos (x) }$. But ...
2
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1answer
52 views

How to split D to calculate $\int_Df(x,y)d(x,y)$ for $f(x,y)=xe^{x^2+y^2}$

Let $$D=\{(x,y)\in \mathbb{R}^2: 1\le x^2+y^2 \le 4 ,\quad y\ge0\}$$ Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined as $$f(x,y)=xe^{x^2+y^2}$$ Calculate $\int_Df(x,y)d(x,y)$ I believe the ...
1
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1answer
88 views

Integrating $f(x) = \frac{1}{x}$ over $[-1,1]$

I am reading a book where it says that improper integral $$\int_{-1}^{1}\frac{1}{x}\,dx$$ is undefined because $$\lim_{b \to 0^-}\int_{-1}^{b}\frac{1}{x}\,dx + \lim_{b \to 0^+}\int_{b}^{1}\frac{1}{x}\,...
13
votes
2answers
195 views

A particular vanishing integral

While dealing with a definite integral on AoPS I discovered (I have to admit by pure chance) the following relation $$\int_0^1\log\left(\frac{(x+1)(x+2)}{x+3}\right)\frac{\mathrm dx}{1+x}~=~0\tag1$$...
0
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3answers
45 views

Is $2\int\frac{d^2x}{dt^2}*\frac{dx}{dt}dt=(\frac{dx}{dt})^2 $ true??

I was watching a video online about motion under inverse square law here and the producer mentioned that, $$2\int\frac{d^2x}{dt^2}*\frac{dx}{dt}dt=\left(\frac{dx}{dt}\right)^2 $$ i donot understand ...
0
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1answer
47 views

Parametric equation and limits

I have a fundamental doubt .Suppose , we are integrating a double integral in X-Y plane , with parametric equation reducing to $$ r= a \cos{t} $$.Then , I am of the view that limits of t should ...
2
votes
5answers
84 views

Integrating $\int \frac{-\sin x}{1+\cos x}\, dx$, I get $\ln(1 + \cos x)$. WolframAlpha gives $2 \ln(\cos \frac x 2)$. Is WA wrong?

So, I'm watching a tutorial on differential equations, where I encountered this little trick: $$\int \frac{y'}{y}\, dx = \ln(y)$$ It seems perfectly logical and easy to justify, but something fishy ...
0
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0answers
40 views

Easier way to solve the integration

I often need to solve a type of integral which is : $$\int x \ \sin ( n_1 x) \ \sin (n_2 x) \ dx $$ $n_1$ and $n_2$ are integers. where the $n_1 $ and $n_2$ are the integer number. The way I ...
1
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2answers
60 views

Why is $f(x)=\int_{1}^{x^2} \frac{\ln(xt)}{1+t}dt$ continuously differentiable and what is it's derivative?

In doing some old exam questions, I came across the following problem. Let $f:(1,\infty)\rightarrow \mathbb{R}$ $$f(x)=\int_{1}^{x^2} \frac{\ln(xt)}{1+t}dt$$ Questions: a) Reason as to why f is ...
2
votes
4answers
103 views

Integrating $\int \frac{dx}{\sqrt{x} (1 + x^2)}$

I'm trying to find the antiderivative of $$ I = \int \frac{dx}{\sqrt{x} (1 + x^2)}$$ but I've been stuck on it for a while. (I came across it in this Youtube video). I know the antiderivative of $\...
2
votes
1answer
327 views
0
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1answer
51 views

Evaluating $\int_{0}^{1}\sqrt{\frac{x}{1-x}}\mathrm{d}x$ [duplicate]

$\int_0^{1}\sqrt{\frac{x}{1-x}}\mathrm{d}x=\frac{\pi}{2}$ This integral seems to be an identity, since the antiderivative for $\sqrt{\frac{x}{1-x}}$ is somewhat cumbersome and the integrand has a ...
2
votes
0answers
65 views

Computing $\int_{-1}^1 \frac{1}{x^2} dx$ (improper integral)

$$ \begin{aligned} & \phantom{ {}={} } \int_{-1}^1 \frac{1}{x^2} \, dx \\ &= \int_{-1}^0 \frac{1}{x^2} dx + \int_0^1 \frac{1}{x^2} dx \\ &=\lim_{a\to 0^-} \left[ \int_{-1}^a \frac{1}{x^2} ...
1
vote
1answer
46 views

Seeking Name of Theory for multiple integral

Consider two definite integrals: \begin{equation} I_1 = \int_{R_1} f(x) \:dx\qquad I_2 = \int_{R_2} g(y) \:dy \end{equation} Then, \begin{equation} I_1 \cdot I_2 = \left[ \int_{R_1} f(x) \:dx \...
0
votes
1answer
36 views

Integration Question involving the Divergence Theorem

I'm doing some practice questions for an upcoming test, but I got stuck on one. It says "Let $\text{div}(F)=x^2+y^2+z^2+3$. Calculate $$\int\int_{S_1}f\cdot\textbf{n}\ dA,$$ where $S_1$ is the sphere ...
1
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0answers
46 views

Searching for closed-form solutions to integral of dilogarithm

while trying to evaluate an infinite sum, I came across this integral: $$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\...
-2
votes
0answers
52 views

Calculate $\frac{\Gamma(N/2)}{\Gamma(N/2+k+\frac 12)}$

Let $N, k\in N$. How to evaluate $$ \frac{\Gamma(N/2)}{\Gamma(N/2+k+\frac 12)} $$ Here $\Gamma$ is the Euler gamma function. All I know is that $$\Gamma(n+1) = n!$$
0
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3answers
70 views

How does $I(x) = \int_x^1 \sqrt{x^{-2}-1} \ dx$ evaluate to $I(x) = -\sqrt{1-x^2}+\ln\bigg[\bigg(1 + \sqrt{1^2-x^2} \bigg)/x \bigg]$?

I came across the following integral $$ I(x) = \int_x^1 \sqrt{x^{-2}-1} \ dx, \quad \quad x < 1, $$ in a paper (see equation (A.6) here) where it is stated that this integral can be evaluated as (...
1
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0answers
30 views

Is every riemann-integral the limit of equidistant riemann-sums?

So, I know that the riemann-integral is sometimes defined as the limit of equidistant riemann-sums like this: which is probably fallacious, as the existence of this limit wouldn't actually imply ...
0
votes
3answers
63 views

Prove that $\int_0^\infty \frac{\sin (tx)(1-\cos x)}{x}dx=\frac{\pi}{2}$ where $0<t<\pi$

Problem : Show that $$\int_0^\infty \frac{\sin (tx)(1-\cos x)}{x}dx=\frac{\pi}{2}$$ Where $0<t<\pi$. At first, I set new parameter $t$. $$f(t)=\int_0^\infty \frac{\sin (tx)(1-\cos x)}{x}dx$$ ...
0
votes
0answers
96 views

Can a non-negative function, zero on the rationals, have a positive integral?

Take a function $cos^2(x)$. It has zeros at $\pi/2 + {\pi}\cdot n$. Then consider the function $$f(x) = (cos^2(x))^{1/x^2}$$ At $x>0$ it also has zeros at $\pi/2 + {\pi}\cdot n$, but as $x \to \...
0
votes
3answers
37 views

What is the total area enclosed between the curve $y=x^2-1$, the x-axis and the lines $x=-2$ and $x=2$?

What is the total area enclosed between the curve $y=x^2-1$, the x-axis and the lines $x=-2$ and $x=2$? I tried to find the area by using the integrals $\int_1^2$ and $\int_{-1}^{-2}$ . $x^2-1$ ...
8
votes
2answers
109 views

How to find the area of a region bounded by a simple closed curve?

I have the following equation: $$ \frac{p}{(a-x)^2+y^2}+\frac{1-p}{(b-x)^2+y^2}=1 \text{ where } 0\leq p\leq 1 $$ Which represent a simple close curve. Obviously, when $p=0,p=1$ or $a=b$ we recover a ...
3
votes
2answers
92 views

Evaluating $\int \sqrt\frac{x^3-3}{x^{11}} dx$

$$\int \sqrt\frac{x^3-3}{x^{11}}\,dx$$ The form the answer takes suggests a very quick substitution should be possible. I cannot see how to obtain it in only a few steps and would be grateful for any ...
0
votes
1answer
58 views

Fourier transform of $\log(i (x-i\epsilon))$

I am trying evaluate $I(\omega) = \lim_{\epsilon \rightarrow 0^+} \int_{-\infty}^{\infty} dx \log(i(x-i\epsilon)) e^{i\omega x}$ The answer for $\omega \in \mathbb{R}$ should be a distribution. My ...
0
votes
1answer
30 views

Volume of two spheres using triple integrals

Lets have two spheres, the middle point of sphere 1 is on the edge of sphere2(see picture). If I want to calculate the volume that is inside this region of the two spheres, do I need to use ...
1
vote
2answers
63 views

Solution to a given differential equation

I am stuck since yesterday to understand how to find the right solution to the differential equation as posted in the image below Then, I tried to solve as shown below, but I am not able to reach a ...
0
votes
2answers
58 views

The integral that $\int_{0}^{\pi/2} \frac{x\sin x+2-2 \cos x}{2\sin x+x\cos x+x}\,\mathrm dx=\ln 2$

While manipulating with some integrals, this integral $$\int_{0}^{\pi/2} \frac{x\sin x+2-2 \cos x}{2\sin x+x\cos x+x}\,\mathrm dx=\ln 2$$ turned out to be doable. So the question is: How to show it ...
-1
votes
0answers
34 views

What is the area? I did not understand the meaning of given ordinate.

What is the area enclosed by the curve $$4y=x^3~~ ,$$ $~x~$ axis and ordinate $~(4,0) ~$?
0
votes
1answer
42 views

Aligning dot product with spherical coordinates for integrals

I am slightly doubting something I have always thought obvious at the moment. Consider two vectors $\vec{a},\vec{b}$ in $\mathbb{R}^3$. We know that $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\gamma$...
3
votes
0answers
44 views

Conditions for interchanging Ito and Riemannian integrals

Let $X_{s,\ r}$ be an adapted real-valued stochastic process, $s,\ r \geq 0$ and $W_s$ be the standard one-dimensional Wiener process (Brownian motion). Under which condition on $X_{s,\ r}$ does the ...
1
vote
1answer
44 views

Identity principle for harmonic functions

The question: Show that if $u$ is real and harmonic on a connected open set $U \subseteq \mathbb{C}$, and $ u = 0$ on some small disc $D(P,r) \subseteq U$, then $u = 0$ on all of U. Now here is a ...
0
votes
1answer
31 views

For which p does $v_p$ have an anti-derivative?

On $\mathbb{R}^2\setminus \{(0,0)\}$ and for real $p$ let $v_p$ be the vector field: $$ v_p(x,y):=\left(\frac{-y}{(x^2+y^2)^p}, \frac{x}{(x^2+y^2)^p}\right)$$ For which $p>0$ is there an open set $...
5
votes
3answers
238 views

Integral $\int_0^1 \frac{\ln(1-x)\ln(1+x^2)}{x}dx$

I am trying to solve by a different approach the fourth sum from here, namely: $$S= \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(4n+m)} =\int_0^1 \frac{\ln(1-x)\ln(1-x^4)}{x}dx= \frac{67}{32} \...
-1
votes
0answers
58 views

Evaluating an algebraic integral

Can somebody provide me with an easy solution.It was supposed to be an high school question , but using online integral it gave me some weird answer . $$ \int \dfrac{3x^3+2x^{11}}{(2x^4+3x^2+11)^4}\;...