1
vote
1answer
115 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
2
votes
2answers
90 views

How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$?

is there a way to explicitely evaluate this integral for natural numbers $n,m$: $$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ Apparently, if $n$ is odd, this integral is zero due to symmetry.
3
votes
1answer
22 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
58 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
3
votes
2answers
243 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
62 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
0
votes
2answers
41 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
3
votes
2answers
133 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 ...
2
votes
4answers
135 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
11
votes
3answers
227 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
0
votes
1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
2
votes
3answers
95 views

Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
1
vote
1answer
42 views

How to evaluate this integral?

How to evaluate $$ \int_{a}^{b} [x] \ dx \ \ + \int_{a}^{b} [-x] \ dx \ ? $$ I know that $[-x] = -[x]$ if $x$ is an integer, whereas $[-x] = -[x] - 1$ if $x$ is not an integer. So is it about ...
1
vote
2answers
107 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
17
votes
1answer
345 views

Integral $\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}$

Please help me prove the following identity: $$\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}\quad a>0$$ This integral is from the ...
2
votes
4answers
129 views

Center of mass of a right circular cone

How can one find the center of a right circular cone with height $h$ and radius $r$? I've found these formulas: $$M_{xy} = \iiint\limits_V z \rho (x,y,z) \, dx \, dy \, dz$$ $$M_{yz} = ...
5
votes
1answer
97 views

How to calculate the integration $\int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2}$ [duplicate]

Given that $$ \int_{0}^{\pi}\frac{dx}{2-\cos{x}}=\frac{\pi}{\sqrt{a^2-1}} $$ How to calculate the integral $$ \int_{0}^{\pi}\frac{dx}{(2-\cos{x})^2} $$
1
vote
1answer
104 views

Double Integral $\iint_D\ (x+2y)\ dxdy$

$$\iint_D (x+2y)\ dxdy $$ If the area is range by $x=2,\ x=3,\ y=x,\ y=2x$, how to include the lines? How limits for integral will looks like? You mean something like this? ( I made mess) $$\iint_D ...
5
votes
3answers
121 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
1
vote
1answer
42 views

Integral Continuation $\Gamma(z)=\int_{0}^{1} e^{-t} t^{z-1} dt +\int_{1}^{\infty} e^{-t} t^{z-1}dt$

I am trying to obtain an analytical continuation for $\Gamma(z)$ into the region of the complex plane characterized by $\Re(z) \leq 0$ but am stuck. Starting from the integral definition of ...
4
votes
4answers
161 views

Integral $P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx$

I am trying to calculate the following principle value integral \begin{equation} P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx \end{equation} for $\lambda \in [0,1].$ I tried to turn this into a ...
15
votes
4answers
313 views

For which $n$ is $ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx}= \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$?

I have been trying to figure out for which $n$ is $$ \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin nx} = \int_0^{\pi/2} \frac{\mathrm{d}x}{2+\sin x}=\frac{\pi}{3\sqrt{3\,}\,}$$ Using maple I got the ...
1
vote
2answers
67 views

Approximate a definite integral to three decimal places: $\int_0^2 \frac{dx}{\sqrt[3]{64+x^3}}$.

I try to expand function $$\frac1{\sqrt[3]{64+x^3}}$$ using Maclaurin series. So, $f(x) = 64{(1+ \frac{x^3}{64})}^{-1/3}$. I expand it and I get ...
5
votes
1answer
91 views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$

I'm trying hard to solve this integral but I still don't know how... $$\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$$ The integral from $-\infty$ to $\infty$ is quite easy, but how could we integrate ...
11
votes
4answers
367 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
1
vote
1answer
76 views

double integral.

I just received this problems from a friend, and I think its a HW problem. its: $$ \int_1^e \int_{1+y^2}^5 \cos (x- \ln x) \ dx \ dy $$ I looked at it, and If I did graph the the region right then ...
0
votes
1answer
46 views

Find the area of a quadrangle with green´s theorem

How can i prove using GreenĀ“s Theorem that the area of a quadrangle with coordinates $(x_i,y_i)$ , $i=1,...,4$ is: $$A = \sum_{i=1}^{4} (x_{i}y_{i+1}-x_{i+1}y_{i})$$ I prove this with some examples ...
3
votes
1answer
211 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
2
votes
2answers
65 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
1
vote
4answers
41 views

verifying log rules via elementary properties of an integral

my attempt at this question so far, $$\int_1^x \frac{1}{t}dt +\int_1^y \frac{1}{t}dt= 2\int_1^x \frac{1}{t}dt+ \int_x^y \frac{1}{t}dt$$ But I am not sure hwo to prove it from elementary ...
3
votes
0answers
43 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
7
votes
3answers
739 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
0
votes
1answer
95 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
0
votes
1answer
37 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
2
votes
1answer
45 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
1
vote
1answer
34 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
2
votes
4answers
102 views

Solving the integral $\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$

I really don't know how to solve this integral $$\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$$ Should I use firstly a formula of $\sin(a+b)$?
13
votes
1answer
257 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
8
votes
1answer
173 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
0
votes
1answer
44 views

Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
4
votes
0answers
168 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
6
votes
1answer
67 views

Integral $\int_0^\infty \frac{1}{(1+x^m)(1+x^2)}\,dx$ [duplicate]

I saw somewhere that the above integral is equal to $\pi/4$ for all real number $m$. This seems to be surprising. Does anyone have a nice proof?
6
votes
3answers
303 views

Evaluating $\int_{0}^{1}\frac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$

Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$ My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found ...
0
votes
4answers
129 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
0
votes
1answer
56 views

Validity of an integral inequality

Suppose we have two functions $f(x)$ and $g(x)$ And $f(x)<g(x)$ for all values of $n$ Then for arbitrary $a$ and $b$ (within the range) is it true that $$\int_b^{a}\frac{dx}{f(x)} > ...
0
votes
0answers
37 views

Computing an (iterated) integral

I am having problems to find a closed form depending on $n$ of the following integral. $$ \int_{t_{0}\ <\ w_{1}\ <\ \cdots\ <\ w_{n}\ <\ t}\ {{\rm d}w_{1}\cdots {\rm d}w_{n} \over ...
1
vote
1answer
50 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
8
votes
1answer
127 views

Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...