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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48 views

An equivalent for $\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^n\;dx$

Set $$ I_n :=\int_0^1\left(\frac{1}{\log x} + \frac{1}{1-x}\right)^n \:\mathrm{d}x \qquad n=1,2,3,.... $$ We have $$I_1 =\gamma, \quad I_2 =\log (2 \pi) - \frac 32, \quad I_3 = 6 \log A - ...
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1answer
23 views

Set function integal

We have a vector $y$ ($\sum_i y_i=1$). Define $S(r) = \{i, y_i\geq r \}$. Here is an integral $\int_{0}^{\infty} |S(r)| dr=\sum_i y_i$. I don't know why the integral is correct. Can anybody help me?
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1answer
52 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
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1answer
16 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
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1answer
52 views

Evaluate the area of the region bounded by the ellipse, where is my mistake?

$ (10x^2+6xy+y^2=2)$ => $ ((x/\sqrt2)^{2} + ((3x+y)/\sqrt2))^{2} = 1 $ so if I change the variables to $u$ and $v$, $u = x/\sqrt2$ $v= (3x+y)/\sqrt2) $ Then my bounds of integration become $-1 ...
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25 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
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13 views

How do you obtain the version of Simpson's rule required as well as deduce the composite integration rule? [on hold]

Consider the function $$g(x)=f(a+(x−1)h)$$ and obtain a version of Simpson’s rule applicable to an integral $$\int_{a+h}^{a−h}f(x)dx.$$ Then deduce the composite integration rule ...
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2answers
79 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
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1answer
163 views

How find this integral $\int_{0}^{\infty}\frac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}$

prove that this integral $$\int_{0}^{\infty}\dfrac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}= \dfrac{\pi}{2(1+r+r^3+r^6+r^{10}+\cdots}$$ for this integral,I can't find it.and I don't know how ...
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1answer
26 views

How do you solve the second part of the question where i am required to derive Simpson’s integration rule?

When $v(x) = A + Bx + Cx(x − 1)$ show that $$\int_0^2v(x)dx= 2A + 2B + \frac23.$$ By choosing A,B and C so that $y = v(x)$ fits a given curve $y = g(x)$ at $x = 0$, $x = 1$ and $x = 2$ derive ...
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2answers
47 views

How do i calculate the value of $\int_{0}^{1} \frac{\ln{(1+x)}}{1+x^2}$? [duplicate]

How do i calculate the value of the following integral-- $$I=\int_{0}^{1} \frac{\ln{(1+x)}}{1+x^2}$$ I tried doing substitutions like $1+x=t$ etc. I also tried to use the property ...
3
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1answer
39 views

Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
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1answer
55 views

solving integral with complex analysis

I have problems with understanding of the evaluation of this integral below. It has been a long a time ago since I had complex analysis. where $a = (1-\sqrt y )^2$ and $b = (1+\sqrt y )^2$. Now my ...
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0answers
31 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right)$. Can this be found analytically?
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1answer
29 views

Inequality involving Jensen (Rudin's exercise)

Exercise (Rudin, R&CA, no. 3.25). Suppose $\mu$ is a positive measure on the space $X$ and let $f \colon X \to (0,+\infty)$ be such that $\int_X f \, d\mu=1$. Then for every $E \subset X$ ...
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2answers
65 views

How do I find this integral $\int\frac{dx}{2x^4+2x^2-1}$

How I evaluate the above integral? $$\displaystyle\int\dfrac{dx}{2x^4+2x^2-1}$$ I have unsuccessfully tried it more than once. Is there a small substitution that I am missing? And is there any ...
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1answer
26 views

What function to use to get geometric mean in trapezoidal rule?

When deriving a trapezoidal rule an integral of $f(x)$ is switched to integral of new function $g(x)$ approximating the first one $$\int_a^b {f(x)dx}\approx \int_a^b {g(x)dx}$$ where $g(x)$ is a ...
4
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1answer
111 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
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1answer
38 views

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and some condition.

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and ere exists an increasing function $g:[0,1]\rightarrow \mathbb{R}$ such that $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
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1answer
22 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
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1answer
42 views

Complex integration with complex integrands [on hold]

How to solve $$ \int_0^{1+i}(x-y+ix^2)dz$$
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1answer
18 views

$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$

i'm trying to prove the following equality $$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$$ I tried to do the following: ...
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1answer
53 views

Help with double integral

I need to prove if this integral exist (and some others) but i would like to know if there is a condition to say if the integral exist (for example in this case) that would help me solve this kind of ...
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2answers
48 views

Integration question involving Area and f(t)

Well I am doing a question and a link of the image is provided here: I am wondering about my answers for a few parts. $\textbf{Part A:}$ Can you just check if I'm correct on these $$F(0)= 0$$ ...
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3answers
61 views

Find $\frac{dG}{dx}$ of $G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$ [on hold]

Define $$G(x)=\int_0^{x^2}\frac{dt}{t^2+4}.$$ What is $\displaystyle\frac{dG}{dx}$? How do I approach this question? What are the steps? What is the solution?
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1answer
39 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
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1answer
43 views

Existence of a function with certain integral properties

Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that \begin{align*} \int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\ ...
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2answers
19 views

Determining the best way to compute a double integral

The question is: When graphed, this is what it looks like: I thought that the best way to do it would be with respect to y first, then x. The bounds: x/sqrt3 < y < sqrt(4-x^2) 1 < x ...
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3answers
19 views

Definite integral fractional exponent in the denominator

I have come across this question and I cannot understand the step highlighted. I would have expected that the fractional exponents of the terms in the numerator would have a negative value after ...
3
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3answers
60 views

What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $?

Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - ...
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1answer
23 views

Area of a Paraboloid inside a Cylinder

Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.
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2answers
35 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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103 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
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3answers
37 views

Integration Trig Substitution

After making the correct trig substitution what does the integral of $\dfrac{1}{\sqrt{9-x^2}} dx$ reduce to without solving the equation? I reduced it down to the integral of ...
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1answer
95 views

Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ [duplicate]

Evaluation of $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx = \int ...
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2answers
19 views

Question on Green's Theorem

Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$. Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where ...
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0answers
43 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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0answers
56 views

Bound for this integral

Using the fact that $$\sqrt{(1+y^2)} - \sqrt{(1+x^2)} \geq \frac{x}{\sqrt{1+x^2}}(y-x)$$ for each $x,y\in \mathbb{R}$. We need to show that $$L(k)- L(h) \geq \int_a^b \frac{h'}{\sqrt{1+{h'}^2}} ...
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0answers
34 views

Evaluating an improper integral with limits $_{-\infty}^\infty$

When evaluating an improper integral with limits $_{-\infty}^\infty$, why do we need to separate the integral into $\int\limits_a^{\infty} \text{ and } \int\limits_{-\infty}^a$? My homework asked ...
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2answers
68 views

Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
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1answer
20 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
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2answers
65 views

True or False? $\int\limits_0^2(x-x^3)dx$ represents the area under the curve $y=x-x^3$ from 0 to 2.

True or False? $\int\limits_0^2(x-x^3)dx$ represents the area under the curve $y=x-x^3$ from 0 to 2. I said true but my textbook says false. Why? (Stewart: Concepts and Contexts p424 q13)
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0answers
16 views

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$ [duplicate]

Let $f\in C[0,1]$. Compute $\lim_{t\rightarrow \infty} \frac{1}{t} \log \int_0^1 \cosh(tf(x)) dx$. Can anyone give me a hint for this type of problem? I don't know where to start. Thank you!
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0answers
18 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
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0answers
60 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
1
vote
1answer
22 views

How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ? I know absolutely continuous means ...
3
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2answers
123 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
1
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0answers
18 views

Limit/Integration in heat equation

While studying heat equation from PDE by L.Evans, I came across the following limit which I'm not able to prove. For $n>=1, \delta >0$ , $lim_{t \to 0+} \;\;{1 \over ...
6
votes
2answers
181 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
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0answers
27 views

Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...