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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21 views

Convergence of integrals if curve parametrisations converge

Let $\boldsymbol{r}:[a,b]\to\mathbb{R}^3$ be the piecewise continuously differentiable parametrisation of a piecewise smooth curve. If, for all $n\in\mathbb{N}$, $\boldsymbol{r}_n:[a,b]\to\mathbb{R}^...
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1answer
41 views

Differential Equations (Proving)

This is the question --> What I have done; $$ dS/dt = kS(N-S) $$ $$ 1/(S(N-S)) = k dt $$ $$ 1/(SN - S^2) = k dt $$ Therefore $$ 1/(S-2S) * ln|SN-S^2| = kt + c $$ A I on the right track? ...
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1answer
70 views

The integral in my book is wrong, can you confirm?

$$\int\sqrt{4z^2 - 4z + 2} dz$$ My book solution: $$ \left( z-1/2 \right) \sqrt{z^2-z+1/2} + 1/4 \ln\left[ z - 1/2 +\sqrt{z^2-z+1/2} \right] $$ My solution: $$ \left( z-1/2 \right) \sqrt{z^2-z+1/2}...
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4answers
57 views

Find $n$ if the area between the curve of $y=x^n$ and the $y$ axis is $3$ times the area between the curve and the $x$-axis

For this question i tried to find the area for red and blue sections, and equate them by red= 3 blue. However, it didnt work out and I got $b-a = 3(b^n - a^n)$ for the outcome. In the book, it ...
3
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2answers
55 views

Flux integral with vector field in spherical coordinates

I have a vector field $\vec{A}$ that is given in spherical coordinates. $$\vec{A}=\frac{1}{r^2}\hat{e}_{r}$$ I need to calculate the flux integral over a unit sphere in origo (radius 1). I cannot use ...
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1answer
109 views

Integral of a small function

How might I go about calculating $$\int_\pi^\infty \frac{\sin x}{x\log x}\text{d}x$$ I honestly don't know where to start.
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1answer
20 views

What are the limits for this triple integral?

This is probably a very easy/silly question, but still I'm not sure about it. I want to calculate the volume of a body bound between the graph of $x^2+y^2-z^4=1$ (what does it look like?) and the ...
3
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3answers
118 views

Evaluate $\int \frac{1}{\sin x+\sec x}\,dx $

Evaluate $$\int \frac{1}{\sin x+\sec x}\,dx $$ Expressing $\sin x$ and $\cos x$ in terms of $\tan\frac{x}{2}$ i.e. putting $\sin x=\dfrac{2t}{1+t^2}$, $\cos x=\dfrac{1-t^2}{1+t^2}$ and hence $dx=\...
4
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6answers
113 views

Integrate: $\int^1_0\frac{r^3}{\sqrt{4+r^2}}dr$

$$\int_0^1\frac{r^3}{\sqrt{4+r^2}}\ \mathrm dr$$ I have attached my work. I am stuck.
2
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1answer
129 views

Expectation defined as Riemann integral

I have a question related to the expectation of a continuous random variable and its Riemann integral definition. Consider a continuous real-valued random variable $X$ defined on the probability space ...
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1answer
53 views

How to calculate$\int_0^\pi e^{-i c (\sin(t) + \cos(t))} \sin(t)\, dt$?

I would like to calculate the following integral $$I=\int_{0}^{\pi} e^{-i c (\sin(t) + \cos(t))} \sin(t) \,dt $$ Here's what I did: We make the change of variables $s=\cos(t)$, so $$I=\int_{-1}^{1} ...
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2answers
46 views

How do I calculate $\int_0^ax^2\sqrt{a^2-x^2}dx$ via substitution?

I have to calculate the integral $$\int_0^ax^2\sqrt{a^2-x^2}dx$$ using solely substitution (no integration by parts). $a$ is a positive constant. I'm confused on how to do this?
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1answer
61 views

complex integration using residue theorem

I have the following integration $$g_n=\frac{(-1)^n}{2\pi} \int_{-\infty} ^ {\infty} (\frac {e^{-itx}}{(it-1)^n})dt $$. I used the residue theorem to find the integration so the result of the ...
2
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1answer
46 views

Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ [closed]

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates: \begin{cases} x=r\cos(\phi),\\ y=r\sin(\...
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1answer
58 views

Integration exercise #1

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a continuous function such that $f(x)\geq 0$ for all $x$ and \begin{align*} \int_{-\infty}^{\infty}f(x) = 1 \end{align*} For $r\geq 0$, let \begin{align*} I_{...
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2answers
83 views

Integrate the square root of tangent [duplicate]

Find $\displaystyle \int \sqrt{\tan(x)}dx$. According to an integral calculator, the answer to this question is $$(-2 \tan^{-1}(1-\sqrt(2) \sqrt(\tan(x)))+2 \tan^{-1}(\sqrt(2) \sqrt(\tan(x))+1)+\log(...
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1answer
36 views

Integrating absolute terms

This is just to clarify my doubt regarding absolute values functions. Lets say there is a function $$f(x) = ax^{2} - \left|\frac{bx}{c}\right|$$ and we are asked to integrate this over $-\infty \to \...
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1answer
49 views

Riemann Integration, question from Munkres

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2\rightarrow$ $\mathbb R$ be defined by setting $f(x,y) = 0$ if $x \not= y$ and $f(x,y) = 1$ if $x = y$. Show that $f$ is integrable over $[0,1]^2$ ...
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0answers
70 views

A modified identity to an integration

$$\int_0^\infty \text{d}x \frac{\sin ax}{e^{2\pi x}-1}=\frac{1}{4}\coth \frac{a}{2}-\frac{1}{2a}$$ Is there a closed form solution to this, slightly different integral (in terms of $a$)? $$\int_{\...
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2answers
424 views

Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of Malmsten—Vardi&...
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1answer
58 views

Show that this integral is finite: $ \int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx $

Haw to prove that the following integral $$ \int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx $$ is finite ? where $a>0$. thanks you in advance
14
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2answers
216 views

Need help with $\int_0^1\frac{\log(1+x)-\log(1-x)}{\left(1+\log^2x\right)x}\,dx$

Please help me to evaluate this integral $$\int_0^1\frac{\log(1+x)-\log(1-x)}{\left(1+\log^2x\right)x}\,dx$$ I tried a change of variable $x=\tanh z$, that transforms it into the form $$\int_0^\infty\...
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3answers
119 views

Properties of improper integral (showing that: $\int \limits_{0}^{\infty}f(x)dx=\int \limits_{0}^{1}f(x)dx+\int \limits_{1}^{\infty}f(x)dx.$)

Let $f(x)$ is integrable on every segment $[r,\infty)$ where $r>0$. Let $\int \limits_{0}^{1}f(x)dx$ and $\int \limits_{1}^{\infty}f(x)dx$ converges. Why in this case we can conclude that $$\int \...
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0answers
31 views

Applicability of monotone convergence theorem, dominated convergence theorem and Fatou's Lemma

In the Monotone Convergence Theorem, the Dominated Convergence Theorem and the Fatou's lemma is having Lebesgue Integrable functions (i.e. functions with finite Lebesgue Integral) a necessary ...
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1answer
47 views

Existence Lebesgue integral and Lebesgue integrability of a function

I have a question related to the existence of Lebesgue Integral. Here in the paragraph "signed function", we read that the Lebesgue integral exists provided that $$(1) \min(\int_{E}f^+d\mu, \int_Rf^{...
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4answers
79 views

Evaluate:$\int_{l}(z^2+\bar{z}z)dz$

Evaluate: $\displaystyle=\int_{l}(z^2+\bar{z}z)dz\,\,\,\,\,\,\,\,\,\,l:|z|=1,0\leq\arg z\leq\pi$ My try: $$\int_{l}(z^2+\bar{z}z)dz=\int_{0}^{\pi}(r^2e^{2i\theta}+re^{i\theta}???)dz$$ I'm stuck ...
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3answers
40 views

how do I proceed further in this integration?

Question : let $n$ be a positive integer. For a real number $x$ , let $[x]$ denote the largest integer not exceeding $x$ and $F=x-[x]$ then I have to prove $$\int_1^{n+1}\frac{F^{[x]}}{[x]}dx=\dfrac{...
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1answer
19 views

Evaluate:$\int_{l}(4z^2-2iz)dz$

Evaluate: $\displaystyle=\int_{l}(4z^2-2iz)dz\,\,\,\,\,\,\,\,\,\,l:y=x^3+2x^2-2x$ between the points $(0,0)$ to $(1,1)$ My try: $$\int_{l}(4z^2-2iz)dz=\int_{0}^{1+i}(4z^2-2iz)dz=\frac{4z^3}{3}-\...
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0answers
20 views

Clarification on the existence of integrals and notation

Consider the random variable $X: \Omega\rightarrow \mathbb{R}^l$ defined on the probability space $(\Omega, \mathcal{A}, P)$, with image $\mathcal{X}\subseteq \mathbb{R}^l$. Consider the measurable ...
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2answers
52 views

Monotone convergence theorem without monotonicity

I have a question related to the monotone convergence theorem. Consider a situation in which all assumptions of the monotone convergence theorem are satisfied except the monotonicity, i.e. we have a ...
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2answers
75 views

Why cannot a primitive of $\frac{1}{1+x^2}$ be equal to $-\textrm{arccot} x$? [closed]

Just giving a thought as to why $\int\frac1{1+x^2}\, dx$ cannot be $-\operatorname{arccot} (x)$?
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1answer
29 views

Integration (Surface Area)

This is the question --> What I have done $$ S = 2π \int_a^b f(x)\sqrt{1+[f'(x)]^2} dx$$ $as$ $$ f(x) = x^3 + {1\over 12x} $$ $therefore$ $$ f'(x) = 3x^2 - {1\over 12x^2}$$ $$ [f'(x)]^2 = {...
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0answers
41 views

Kinetic Energy of a rotating cone

We have a regular circular cone of base radius R, height H and equally distributed mass M that rotates on its axis with a circular velocity $\omega$. How do we calculate it's energy via integration? ...
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0answers
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Confusion regarding bounds on integral

So i have presently seperated a differential equation to shown that $\frac{dp(y)}{p(y)}=-\frac{Mg}{RT(y)}dy$ I would then like to integrate on both sides to solve for $p(y)$, but i am unsure of what ...
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2answers
61 views

finding an integral .

I can’t find the integral $$\int x^2 \sqrt{2x-4}dx, $$ and am so confused between using the integration by parts or the integration by u substitution, if any one please can tell me which method to ...
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3answers
110 views

How to solve $y'=e^{\frac{xy'}y}$?

How to solve the following equation? $$y'=e^{\frac{xy'}y}$$ We must find a common solution.
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3answers
417 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable $...
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1answer
65 views

Area under the curve (Integration)

For this question: I have found the limits to be $(-k)^{\frac{1}{2}} , 0 , k^{\frac{1}{2}}$ but the trouble I'm having is how do I know what graph I should minus from each other. So for area a , ...
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4answers
101 views

Integral Property: $\int^a_{0}f(x)dx=\int^a_{0}f(a-x)dx$ [Proof by definition of Riemann Sums]

This link: Why $\int_0^af(x)dx=\int_0^af(a-x)dx$? addresses this question but I do not follow the proofs in the answers: Each proof starts off with variable $x$ but ends the right hand side with a ...
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0answers
63 views

Integrate $\int^1_{0} \frac{\ln (x+1)}{x^2+1}dx$ [duplicate]

$$\int^1_{0}\frac{\ln (x+1)}{x^2+1}dx$$ I'm having trouble solving this one. I tried trigonometric subst. but that doesn't get me far:$$\int^1_{0}\frac{\ ln (tan\theta+1)}{\sec^2\theta}\sec^2\theta ...
2
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1answer
68 views

Integral for Biot-Savart

What strategy is the quickest for solving the following integral? Note: this integral is generated by the need to determine the magnetic field at a point along the z-axis generated by a wire of length ...
4
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2answers
127 views

Find $\int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx$ [closed]

Find $\displaystyle \int_{0}^{\infty} \dfrac{\ln(x)}{1+x^2}dx$. How should I change the limits of integration to evaluate this?
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1answer
146 views

Integrating $\sin(x)/x$, how to treat the pole at the origin? [duplicate]

I want to use residue theory to integrate $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ What would be a good contour to use? I plan to take the imaginary part of this integral: $$\int \frac {e^{...
2
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2answers
51 views

Evaluate the integral $\int_0^n \frac{f(x)}{f(x) + f(n-x)}dx $

I found this problem on an old FB post and have just been humbled by it. Let $f(x)$ be continuous on $[0,n]$ such that $n > 0$ and $f(x) + f(n - x)$ does not vanish on $[0,n]$ then evaluate $$ \...
3
votes
1answer
38 views

Integration involving Inner Product

Suppose $f: {\bf R}^n \to {\bf R}^n $be a continuous function such that $\int_{{\bf R}^n} \vert f(x) \vert \, dx < \infty$. Let $ A \in GL_n({\bf R})$. Show that $$ \int_{{\bf R}^n} f(Ax) e^{i\...
2
votes
0answers
48 views

Calculate the line integral $ \oint\limits_C e^{x+y} \, dy$

Calculate the line integral $ \oint\limits_C e^{x+y} \, dy$ over the part of lemniscate: $ (x^2 + y^2)^2 = x^2 - y^2, x \ge 0 $ I tryed the straight way: $ x = r \cos \phi, y = r \sin \phi $. Then ...
0
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1answer
35 views

Help getting this integral into specific form for integration.(Polynomial Division)

I need to get $$\frac{x^4-2x^2+4x+1}{x^3-x^2-x+1} $$ into a specific form for integration. I have factored out my denominator to $(x-1)^2(x+1)$ through grouping. I am unsure how to use polynomial ...
0
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1answer
68 views

Integrate this Spherical Harmonics Function [closed]

I am interested in the following integral $$\int_0^{2\pi}\int_0^\pi\mathop{\mathrm{d}\theta}\mathop{\mathrm{d}\phi} \sin\theta Y_l^{m*}(\theta,\phi)Y_{l'}^{m'}(\theta, \phi)\cos^2\theta\cos^2\phi,$$ ...
2
votes
2answers
90 views

Find $\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\tan(x)^{\sqrt{2}}}dx$. [duplicate]

Find $\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{1}{1+\tan(x)^{\sqrt{2}}}dx$. I am told I should solve this by switching the limits, but I am unsure of the substitution I should use.
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4answers
185 views

Evaluating the Definite Integral $\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$

How can I find this integral $$I=\int_{0}^{1}\frac{2 \sin \pi x \cos \pi x}{1+x^2}dx$$ Any trick that could compute the definite integral is acceptable. However, it will be more challenging to ...