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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1answer
46 views

Using trig to integrate $x^2/\sqrt{16-x^2}$

I'm trying to integrate: $\int\frac{x^2}{\sqrt{16-x^2}}\ dx$ My try was to convert $x$ to $4\sin(u)$ and $dx$ to $4\cos(u)du$, but I'm not sure. Thanks.
1
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0answers
30 views

Integral equations iterates question

Suppose that $K(x,y) = g(x)h(y)$ and that $\int_{a}^{b} g(x)h(x) dx=0$. Let $\psi_0(x)=f(x)$. Show that all iterates equal the first iterate and find a simple formula for the solution. I'm really ...
4
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2answers
106 views

To prove the integral inequality $\int_\overline{\theta}^{\pi}\frac{d\theta}{\sqrt{1-\lambda \cos{\theta}}}\gt\pi$

The following inequality comes up in connection with motion in a dipole field. One has to show that $$\int_\overline{\theta}^{\pi}\frac{d\theta}{\sqrt{1-\lambda \cos{\theta}}}\gt\pi$$ where ...
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1answer
42 views

Proving that the solution to $yu_x-xu_y=0$ containing $x^2+y^2=a^2,u=y$ doesn't exist.

This is basically a Cauchy problem.Parametrizing the given curve(Initial curve):$x=a\cos s,y=a\sin s,u=a\sin s$ $y'(s)y(s)+x(s)x'(s)=a\cos s(a\sin s)-(a\cos s)(a\sin s)=0 \implies$ Characteristic ...
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1answer
26 views

Path Integral over circle always $0$

Let's say we want to evaluate the integral $$ \int_{\lvert x \rvert = R} f(x) dx $$ where $R \gt 0$ is the radius of a circle. Now we parameterize $\varphi(t) : [0, 2\pi] \rightarrow D $ : $$ ...
1
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2answers
68 views

Integrating $\sqrt{x^2+a^2}$

I'm trying to integrate this function wrt $x$, substituting $x = a \tan \theta$ $$ \int \sqrt{x^2+a^2} dx = a^2 \int \frac {d\theta}{\cos^3\theta} = $$ $$= a^2 \cdot \frac 12 \left( \tan\theta ...
1
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1answer
29 views

Trapezium rule: undefined values of $f(x)$

I am trying to estimate the area between the curve $f(x) = \frac {\sin(x)}{x^2+2x}$ and the $x$ axis between $x=-1$ and $x=2$ using the trapezium with $6$ strips. However, when calculating values for ...
2
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1answer
45 views

Calculate surface area of flat figure by using double integral and polar coordinates

Check me please. I tried check it via WolframAlpha, but I don't trust in it 100%. Task: Calculate surface area of flat figure by using double integral in polar coordinates. Figure confined by line: ...
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2answers
92 views

Evaluating the indefinite integral $\int \tan \sqrt {x} \,dx$

$$\int \tan \sqrt {x} \,dx$$ I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly.
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3answers
138 views

Evaluating:$\sum_{n=0}^\infty\frac1{\binom{2n}{n}}$ [closed]

How to evaluate: $$\sum_{n=0}^\infty\frac1{\binom{2n}{n}}$$ $\binom{n}{r}$ is the binomial coefficient. If possible, present different methods as well.
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1answer
39 views

To test the convergence of $\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$

To test the convergence of the improper integral: $$\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$$ Here we see that $0$ is the point of infinite discontinuty for $p<0$. Let $f(x) = \frac{x^p \log ...
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3answers
72 views

How to compute $\int {\frac{1}{{4 - 9{x^2}}}dx} $? [duplicate]

How can I evaluate the following integral $$\int {\frac{1}{{4 - 9{x^2}}}dx} $$
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1answer
28 views

Solving Differential Equation about rate of infected computers

I am having some trouble solving this differential equation for the rate of infected computers in a botnet at time t $$\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{c\nu (1-x) + \beta x(1-x) - \gamma ...
3
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2answers
30 views

Evaluate the inward flux of the vector field $F=<y,-x,z>$ over the surface $S$ of the solid bounded by $z=\sqrt{x^2+y^2}$ and $z=3$.

Evaluate the inward flux of the vector field $F=<y,-x,z>$ over the surface $S$ of the solid bounded by $z=\sqrt{x^2+y^2}$ and $z=3$. this is basically an inverted cone (right?) So by changing ...
2
votes
3answers
42 views

Surface area of $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$

Let $S$ be the portion of the sphere $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$. Calculate the surface area of $S$ Ok i'm really confused with this one. I know i have to apply the ...
7
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1answer
71 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
1
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1answer
27 views

Question about Integral with exponential function

Please refer to the image below. I would like to ask why the highlighted part would be gone in step $2$ ? What calculation involved making it $0$ ? Thank you so much!
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2answers
58 views

What value of $N$ to use in Simpson's rule to reach desired accuracy?

I need to calculate Simpsons rule for the integral of $$\frac{e^x-1}{\sin x}$$ from $0$ to $\pi/2$ with minimum number of intervals $N$ up to $10^{-6}$ accuracy. Wolfram alpha seems to be giving me a ...
0
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1answer
42 views

What is my mistake: Asymptotic behaviour of the following integral?

Okay, I am going to be very specific. I have the following integral $$\int_{-1}^1 \mathop{dx}\frac{x^{n-2m}(a^2+x^2)^{(k-2)n/2+(3-k)m/2}}{(c_1 ...
1
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1answer
28 views

Definition and existence of Riemann integral in PMA Rudin

I understand all moments bseides $(1)$ and $(2)$. Why Rudin considers $\inf U(P,f)$ and $\sup L(P,f)$. Why we can't considered $\sup U(P,f)$ and $\inf L(P,f)$? Can anyone explain it to me please?
2
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1answer
70 views

Proving $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ and $\int_0^{\infty} x^m (\log x)^m dx$

Prove that: (a) $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ is convergent if $0<m<1$ and $0<n<1$; (b) $\int_0^{\infty} x^m (\log x)^n dx$ is convergent if $m< -1$; Getting no ...
1
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2answers
56 views

Exact value of integral $\int_{0}^{\pi/4}(\sec x-x)(\sec x+x)dx$

In terms of integration, how would you obtain the "exact-value" of $$\int_{0}^\frac\pi4(\sec x-x)(\sec x+x)dx.$$ Note: $1+\tan^2x=\sec^2x$
1
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1answer
19 views

estimates on an improper integral associated with normal distributuion

Show that $\int_{x}^{\infty}e^{-\frac{t^2}{2}}dt\geq e^{-\frac{t^2}{2}}(\frac{1}{x}-\frac{1}{x^3}) $ for all positive $x$ Does it require the mean value theorem, or the Taylor series expansion? It is ...
1
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2answers
36 views

Calculate double integral using Polar coordinate system

Need to calculate $\int_{0}^{R}dx\int_{-\sqrt{{R}^{2}-{x}^{2}}}^{\sqrt{{R}^{2}-{x}^{2}}}cos({x}^{2}+{y}^{2})dy$ My steps: Domain of integration is the circle with center (0,0) and radius R; $x = ...
0
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0answers
17 views

Integral alternative to $\text{Ei}(x)$ function.

Is there an alternative but simple way to find the integral of the following function. I tried to find by the method of "Integration by parts" but it again and again result the answer same as ...
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1answer
17 views

Evaluating a contour-integral.

Consider the ellipse $C$ given by $x^2 + y^2/4 = 1$. How to evaluate $$\int_C x^2 \, \nu(d(x,y))$$ where $\nu$ is the Lebesgue length measure on $C$? I am not sure if this can be computed like a ...
2
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3answers
141 views

Integrate $\sin(\cos(x))$ with respect to $x$ [duplicate]

Solve this: $$\int\sin(\cos x)dx$$ I checked on Maxima, mathematica but both cannot find its integral though numerical approximation is available in later. Has someone faced similar problem? ...
1
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1answer
48 views

How to show continuously differentiable.

Let $$f(x)=\int_{0}^{\infty} e^{-xt} t^x\ \mathsf dt$$ for $x>0$. Show that $f$ is well defined and continuously differentiable on $(0, \infty)$ and compute its derivatives. My confusion is ...
0
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1answer
52 views

Integral inequalities.

If f(t) is non-negative and bounded function, then by Schwartz inequality, we have $(\int\limits_0^tf(s)ds)^2 \leq t\int\limits_0^tf^2(s)ds$. Now my questions are, (i) is there any possibility to ...
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0answers
8 views

using Green's Theorem to calculate the Work done for a vector function.

$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ is a vector. Q is a square $$[-1,1]\times [-1,1]$$ and R also a square $$[1,2]\times [-1,1]$$ How do i calculate the Work Integral about Q and R? of the ...
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1answer
29 views

About the convergence of the improper integrals:

About the convergence of the improper integrals: 1: $\int_0^{\pi/2} \frac{1}{e^x - \cos x} dx$ 2: $\int_0^{\pi} \frac{1}{\cos \alpha - \cos x} dx, 0 \leq \alpha \leq \pi.$ In the first problem $0$ ...
2
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1answer
28 views

Indefinite integral problem: $\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ [duplicate]

$ I =\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ my attempt: $I=\int_1^n \frac{(x - [x])^{[x]}{[x]}}\implies\sum_{i=1}^n\int_r^{r+1} \frac{(x-r)^r}{r}dx $ Now by ...
2
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1answer
60 views

Evaluating $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-\cdots -x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$

I have reduced a problem to evaluating the integral $\int_0^u\int_0^{2u-x_1}\cdots \int_0^{ku-x_1-x_2-...-x_{n-1}}dx_ndx_{n-1}\cdots dx_2dx_1$. I tried computing this for small $n=1,2,3$ but I can't ...
3
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3answers
90 views

Evaluate $\int \frac{dx}{e^{2x}+1}$

Evaluate $$\int \frac{dx}{e^{2x}+1}$$ Here's what I did: $$t=e^{2x} \Rightarrow \frac{1}{2}\int \frac{dt}{t(t+1)} = \frac{1}{2}\left(\int \frac{dt}{t} - \int \frac{dt}{t+1}\right) = ...
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0answers
25 views

Curl and Potential of Vectorfields.

(-y,x) is a vector$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ After using the curl or rotation formula i get: for $a = 0$ and $a = 1$ there is constant curl or rotation. For the latter the Curl is 0 ...
1
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1answer
39 views

What is $\int_Byd(x,y)?$

Let $T=\{(x,y)\in\mathbb{R}^2:x,y\le 1,\; -x^2\le y\le 1-x,\; y\ge 0,\; -y^2\le x\le 1-y,\; x\ge 0\}$. My question: How to determine $$\int_Byd(x,y)?$$ I would start with ...
2
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2answers
59 views

Primitive of an $L^1$ function is continuous

The primitive of a continuous function on a compact interval is continuous via the Fundamental Theorem of Calculus. Let $I \subset \mathbb{R}$ be open and let $u': \overline{I} \mapsto \mathbb{R}$ be ...
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1answer
23 views

How do I continue this surface integral?

I tried to solve this. After find normal vector, I don't know how know how do I continue this.. Question is in this pic. =>]1
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1answer
83 views

how do I find integrals where one limit tends to infinity?

As in this question, where putting in the value of infinity makes it unsolvable. $$\int_0^\infty \frac{1}{\sqrt{x} (x+1)} \, dx$$ so I wrote this integral as an integral with limits 0 to t where t ...
0
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1answer
29 views

Find the rule of $f(x)$ given the following definite integrals

Let $f$ be a differentiable function defined for $x>2$, where $a>1$ and $b>1$, such that: $$\int_{3}^{ab+2}{f\left( x \right)dx=\int_{3}^{a+2}{f\left( x \right)dx+\int_{3}^{b+2}{f\left( x ...
2
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1answer
67 views

Evaluate the following definite integral?

How to evaluate the following integral?
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1answer
36 views

To examine the convergence of the improper integrals

To examine the convergence of the improper integrals: 1: $\int_0^{\infty} \frac1{x \log x}dx$ 2: $\int_0^{\infty} \frac1{(x +sin^2x) \log x}dx$ In both the cases we see that $0$ and $1$ are point ...
1
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1answer
35 views

Is this the only possibility for $F(x)$

We're given that $\int_{0}^{a}{f\left( x \right)dx=a}$ and $f(x)$ is continuous, then: $$\int_{0}^{a}{f\left( x \right)dx=a}=F\left( a \right)-F\left( 0 \right)$$ $$F\left( a \right)-F\left( 0 ...
2
votes
2answers
151 views

How to find the indefinite integral for a natural log being divided by x?

I've done many examples in the math book but none of them have a natural log as a numerator. Here's the question. $$\int\frac{(\ln\ x)^7}{x} dx$$ I am given these 2 properties, where $u$ is a ...
2
votes
2answers
150 views

Why doesn't exist a Cousin's lemma for left-tagged partitions?

I am thinking of the possible validity of a statement like this: given any positive mapping $\delta$ on $[a,b]$, there exists a partition $\, a=a_0<a_1<\cdots<a_n=b \,$ of $ \,[a,b] \,$ so ...
0
votes
0answers
29 views

Finding moment of intertia using spherical coordinates

I have taken a picture of the problem with my work. The answer I am getting is incorrect. I think I am doing something wrong after the following. Any help is appreciated `
3
votes
1answer
91 views

Evaluation of $\int_{0}^{1}\frac{x^{2015}-1}{\ln x}dx$

Evaluation of $\displaystyle \int_{0}^{1}\frac{x^{2015}-1}{\ln x}dx\;\;$ $\bf{My\; Try::}$ Let $$I(a) = \int_{0}^{1}\frac{x^{a}-1}{\ln x}dx\;,$$ Then $$I'(a) = \int_{0}^{1}\frac{x^a\cdot ...
4
votes
1answer
204 views

Evaluate $\int_{-1/2}^{1/2}\frac{\sin^4(n\pi f)}{\lvert\sin(\pi f\lvert^{2d}[\sin(\pi f)]^{2}}df$ for any $d\in(-1/2,1]$

Evaluate $$\int_{-1/2}^{1/2}\frac{\sin^4(n \,\pi f)}{\lvert\sin(\pi f)\lvert^{2d}\sin(\pi f)^{2}}df$$ for any $d\in(-1/2,1]$ This was helpful, but what if the denominator is raised to a fractional ...
3
votes
2answers
84 views

Differential Notation Magic in Integration by u-Substitution [duplicate]

I'm really confused now. I always thought that the differential notation $\frac{df}{dx}$ was just that, a notation. But somehow when doing integration by u-substitution I'm told that you can turn ...
1
vote
4answers
127 views

Find the indefinite integral $\int {dx \over {(1+x^2) \sqrt{1-x^2}}} $

Find the indefinite integral $$\int {dx \over {(1+x^2) \sqrt{1-x^2}}} $$ Is there a smart substitution or algebric trick that I'm missing? Because integration by parts hasn't helped..