All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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2
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0answers
29 views

Help changing the order of integration

So I need to change the order of integration. I am giving the following limits, $1 \leq x \leq 9$ and $\sqrt{x} \leq y \leq 4$. I am having no luck solving this one. Any help would be greatly ...
1
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0answers
23 views

Double integral help

I'm having difficulty with a question. It says By putting $x=r\cos(\theta), y=r\sin(\theta)$, prove that $$\int_0^{\infty}\int_0^{\infty}e^{-(x^2 + 2xy\cos(\alpha)+y^2)}dx\ ...
-1
votes
0answers
9 views

Integral Evaluation with MATLAB-Mupad (triple and lesser degree integrals)

https://www.wolframalpha.com/input/?i=integral+of+2c%28x%5E2%2By%5E2%29%28√%28a%5E2+-+x%5E2+-+y%5E2%29%29+with+respect+to+y+from+-√%28a%5E2+-+x%5E2%29++to+√%28a%5E2+-+x%5E2%29 Here is a link to the ...
3
votes
0answers
68 views

The long Integral with a nice result

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2Li_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) -\frac{\zeta(2)-2\log^2 ...
1
vote
2answers
73 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
1
vote
1answer
30 views

Does it Make Sense to Use the Variable of Integration as a Bound?

I can't for the life of me seem to decide if using the variable of integration as a bound makes sense. For instance, integrating $y=x$ from $0$ to $x$. I don't think it does… But I'm not sure.
0
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1answer
15 views

Numerical integration: Quadrature method which one to use?

Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.
1
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1answer
34 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
0
votes
1answer
18 views

natural log integral question dx/(13-x)

Just wondering why the answer to the integral: $$\int \frac{\mathrm{d}x}{13-x}$$ is $-\ln|x-13|$ as opposed to $-\ln|13-x|$. Why do the $13$ and $x$ get switched?
0
votes
1answer
28 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
0
votes
1answer
16 views

Solving a double integral using substitution

The problem: Evaluate $$\iint_{D}(x+y)^2(x-y)^5\:\mathrm{d}x\:\mathrm{d}y,$$ where $D$ is a rectangle with vertices in $(0, 1), (1, 0), (1, 2), (2, 1)$. So I drew the square and thought up this ...
1
vote
1answer
10 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
2
votes
2answers
51 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
3
votes
1answer
57 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
1
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0answers
21 views

Infinitesimal multiplication

Taking the exponential of the sum of logarithms gets you the product of the terms. Now, if we have a function with positive values, we can form $$\exp\int_a^b \ln f(t) dt.$$ In some sense, this is ...
4
votes
1answer
72 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
1
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0answers
13 views

difference of the values of a function is an integral

This is a very simple quesiton but something I don't understand. From Taylor expansion: $$f(y)-f(x)=f'(x)(y-x)+O((y-x)^2)$$ so, if I just picture that, on the left is the difference between two values ...
3
votes
2answers
44 views

Evaluate $\int\frac{\mathrm{d}x}{2x-4}$

My question is to evaluate: $$\int\frac{\mathrm{d}x}{2x-4}$$ Why is the solution equal to $\frac{1}{2}\ln|x-2|$ as opposed to $\frac{1}{2}\ln|2x-4|$? I understand that if I factor $\frac{1}{2}$ ...
3
votes
2answers
71 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
0
votes
1answer
26 views

Evaluate $\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt$

$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\displaystyle\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt=?$ If the limit exists with l'Hopital i get ...
2
votes
2answers
39 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
5
votes
4answers
534 views

Formula for computing integrals

For computing derivative of a function, we can use the definition of a derivative, i.e. $$\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$ Is there some for computing integrals too?
1
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0answers
38 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
0
votes
1answer
44 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
7
votes
3answers
91 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
2
votes
1answer
50 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
2
votes
1answer
18 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
votes
0answers
25 views

Volume by double integration?

Suppose that $h<a<0$. Show that the volume of the solid bounded by the cylinder $x^2+y^2=a^2$, the plane $z=0$ and the plane $z=x+h$ is $V=\pi a^2h$. I'm having a very hard time with ...
-3
votes
1answer
36 views

Integrate $\cot^2x-\frac{\cos^2x}{\tan^2x}$ [on hold]

Integrate $\int{\cot^2x-\frac{\cos^2x}{\tan^2x}}dx$
-3
votes
1answer
41 views

Integrate $\int^{1}_{0}{\sin^2x}$ [on hold]

What is the value of this integration ? $\int^{1}_{0}{\sin^2x}dx $
1
vote
1answer
27 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
69 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
-1
votes
0answers
23 views

Question concerning the integrability of a function

Let $f: [0,1]^2 \to \mathbb{R}$ be a function such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q} \\ 2y & : x \notin \mathbb{Q} \end{array} ...
2
votes
1answer
42 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
5
votes
2answers
92 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
0
votes
0answers
6 views

Find a map T(D*)=D and Triple integral

D={(x,y,z)| (7x-3y-z)^2 +(-3x+7y-z)^2 +(-x-y+3z)^2<=100} D* = {(u,v,w)|u^2+v^2+w^2<=1} find map T(D*)=D express the triple integral of xy dx dy dz over D as an integral over D* and evaluate
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0answers
40 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
1
vote
2answers
41 views

How to integrate $(x^2 - 1)/(x^2 + 1)$?

I have gone until separating $(x^2 - 1)/(x^2 + 1)$ into $x^2/(x^2 + 1)$ - $1/(x^2 + 1)$. The latter fraction I can substitute by $\tan u$, but how to solve the first fraction and how does it all come ...
1
vote
0answers
28 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
4
votes
0answers
54 views

Calculate the following Integral (Please Help)

Hi, I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me ...
1
vote
2answers
40 views

Aside from this two practical technique to compute any integral, what else? [on hold]

Aside from this two practical technique to compute any integral, what else could called a fundamental method but not approximate method like Riemann Sum? These two method I've been referring to are ...
3
votes
1answer
53 views

Is a probability density function necessarily a $L^2$ function?

If a nonnegative continuous real valued function $f$ is integrable over $\mathbb{R}$ with $$\int_\mathbb{R} f\,\mathrm{d}x = 1,$$ does it hold true $$\int_\mathbb{R} f^2 \,\mathrm{d}x<\infty?$$ ...
0
votes
1answer
46 views

How to integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$?

How do I integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$, where $C_1$ is an arbitrary constant? Is this integral really complex (hard to integrate)? EDIT: This comes from DE: $dy/dx = ...
0
votes
1answer
11 views

Newton-cotes formulas help

I am having a hard time understanding how to use this formula. If given the following problem: Compute ∫ sin x dx using Simpson's rule with 3 points in the range 0 to Pi/2. Do I have to take the ...
1
vote
1answer
17 views

Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
1
vote
1answer
21 views

Calculating the center of mass in spherical coordinates

So normally, to calculate the center of mass you would use a triple integral. In my particular problem, I need to calculate the center of mass of an eight of a sphere where it's density is ...
1
vote
1answer
21 views

Primitive function tricks

Calculation here Questions: how did he get the idea to "split up" $r^2$ into $2r \cdot r/2$? is he doing integration by parts after the second = sign? I can't really follow the algebra here.
0
votes
1answer
26 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
4
votes
0answers
55 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$