Questions about the evaluation of specific definite integrals.

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2
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1answer
119 views

Why do we put absolute brackets for ln?

When writing out the final answer in $\ln$ form, why is it necessary to put absolute brackets? How does it affect the answer? I have this answer of $-3\ln|\frac{3+\sqrt{9-x^2}}{x}|$, but why does it ...
2
votes
2answers
42 views

what is the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$?

I tried evaluating the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$. I keep getting $\ln(2) - \ln(2) = 0$, but my textbook says its $\ln(4)$. I'm not sure what I am doing wrong because when I ...
0
votes
3answers
30 views

How does integrating over absolute values work with definite integrals?

I have $ \int_0^\pi | \sin(x/2) | \, dx $, and according to Wolfram Alpha, the indefinite integral is: $$ -2\cos(x/2)\operatorname{sgn}(\sin(x/2)) + C $$ but the definite integral above evaluates to ...
0
votes
1answer
34 views

$\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ [on hold]

$f$ an integratable function defined in $[a;b] \rightarrow \mathbb{R}$: prove that exists $c \in [a;b]$ that: $\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ and then give an example that might not ...
1
vote
1answer
36 views

How to evaluate $\lim_{n\to\infty}\int_0^n\frac{x^2+a^2}{x^4+b^2x^2+b^4}dx$

Evaluate this limit: $$\lim_{n\to\infty}\int_0^n\dfrac{x^2+a^2}{x^4+b^2x^2+b^4}dx$$ I tried to simplify this fraction. I noticed that $x^4+b^2x^2+b^4$ can be written as $$\dfrac{x^6-b^6}{x^2-b^2}$$ ...
1
vote
3answers
49 views

Evaluate $\lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du$

Let $f(u)$ be a function such that $\lim_{u \rightarrow 0} f(u)=0$ e.g. $f(u)={u}$. How would I evaluate $$ \lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du $$ Is this always equal to zero? My ...
3
votes
2answers
106 views

Find $\int_0^a{f(x)}\, dx$

SMT 2013 Calculus #8: The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, ...
-1
votes
0answers
14 views

Need help to solve double integral exercise

I'm facing problems solving these integrals. I can't reach the result. Could anyone help me? There are two problems with the same integral. Integral: $\iint (y) dx dy $, a) $\{B=(x,y) \in R² | ...
0
votes
3answers
40 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
0
votes
1answer
46 views

$\int_0^{\pi/2}\ln(\sin(x))$?

From this paper: http://math.ucsd.edu/~ebender/20B/7_DefInt.pdf Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
2
votes
7answers
187 views

Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$ But I don't know how to explain it. I understand that the derivative of the equation is a ...
3
votes
2answers
38 views

Real integral getting numerically evalued to complex number

I tried feeding th following integral into WolframAlpha: $$\int_0^2\frac{1}{3\sqrt{x}(\log(x))^{\frac13}},$$ to get an idea of its value. Result: http://bit.ly/strangeint ...
0
votes
1answer
20 views

Finding hypervolume lying between Gaussian function and x-y-z plane over $\mathbb{R}^3$

Define the 3-variable Gaussian function by $G(x,y,z) = e^{-(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})}$. Find the hypervolume lying between this surface and the x-y-z hyperplane, over the ...
0
votes
1answer
19 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
0
votes
1answer
20 views

Integral containing Associated Legendre Polynomials

I need to evaluated the following integral: $\int_0^\pi \sin(x) \cos(x) P_l^m(\cos x) P_k^m(\cos x) \mathrm{d}x$ and I thought since a solution is known to a similar thing $\int_0^\pi \sin(x) ...
0
votes
1answer
23 views

Integral of nonnegative function on plane domain gives a negative result, what is wrong?

Given an area $D: x \ge y, 0 \le x \le 1, y \ge 0$. $$ f(x,y)= \begin{cases} 2, & (x,y) \in D,\\ 0, & \text {others}\end{cases} $$ For this area $D_1: x+y \le 1, 0 \le y \le x$, I'm ...
1
vote
1answer
12 views

Parameter Integral Sin function, Gamma function

Given $$ F: A \in\Bbb{R}\rightarrow\Bbb{R}$$ such that $$F(y)=\int\limits_{0}^{\pi/2} (sinx)^y(cosx)^{1-y} \; dx$$ Prove that $F(1/2)=\frac{1}{\sqrt\pi}(\Gamma(3/4))^2$ and then find the maximum ...
2
votes
1answer
75 views

Prove an integral expression equals $\pi\log 2/2$

How do you prove that: $$3\int_0^1 \frac{\tan^{-1}(x)}{x}-2\int_0^{1/2} \frac{\tan^{-1}(x)}{x}-\int_0^{1/3} \frac{\tan^{-1}(x)}{x}-\frac 12 \int_0^{3/4} \frac{\tan^{-1}(x)}{x}=\frac{\pi\log ...
-1
votes
0answers
43 views

Using Riemann sum to prove that $\int_a^b \sin(x)dx=\cos(a)-\cos(b)$ [on hold]

Using the Riemann Sum, how would you prove the definite integrals of $$\int_a^b\sin(x)dx=\cos(a)-\cos(b)$$
2
votes
0answers
31 views

Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
0
votes
1answer
21 views

Finding the volume using cylindrical shells about the x-axis

So I have spent about a hour on this problem and figured it was time to ask for some advice. The problem is to find the volume using cylindrical shells by rotating the region bounded by $$8y = ...
2
votes
2answers
82 views

Finding limit using Riemann integral

$$\lim _{n\rightarrow \infty }\sum_{i=1}^{n}\frac{n}{\left ( i-1 \right )^{2}+n^{2}}$$ What is the idea behind this? I have watched an MIT open courseware video on this kind of problems, and what I ...
8
votes
4answers
108 views

Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$

Suppose $f$ is a continuous, strictly increasing function defined on a closed interval $[a,b]$ such that $f^{-1}$ is the inverse function of $f$. Prove that, ...
2
votes
2answers
27 views

Evaluating differential entropies with Matlab: NaN issue

With Matlab I am trying to evaluate differential entropies. These are integrals like $$\int_\mathbb{R} p(x) \log (p(x)) \mathrm{d}x$$ where $p(x)$ is a probability density function. My $p(x)$ is ...
1
vote
1answer
56 views

How can I prove this integration result?

The question is: how can I prove that: $$\int_{0}^{\pi} \sin^n\theta\ d\theta = \frac{\Gamma\big(\frac{1}{2}\big) \Gamma\big(\frac{1}{2} + \frac{1}{2}n\big)}{\Gamma\big(1 + \frac{1}{2}n\big)}$$
0
votes
1answer
30 views

what are the properties of the definite integral that are related to inequalities? [on hold]

what are the properties of the definite integral that are related to inequalities? I've been searching the internet and asking teachers regarding this seemingly implausible connection, but haven't ...
1
vote
1answer
37 views

A function such that $f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$ for all $x$

Let $f:\mathbb R\to\mathbb R$ be a function with continuous derivative such that $f(\sqrt{2})=2$ and $$f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$$ for all $x\in\mathbb R$. Find ...
1
vote
1answer
71 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
6
votes
0answers
81 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
1
vote
0answers
29 views

Shell Method About Y-Axis

In my calculus course, we just covered the Shell Method and its uses. I have been doing the homework for a few hours and I am absolutely stumped by a question. The question states: Find the ...
1
vote
2answers
47 views

Triple Integral of $1/\sqrt{2 + x^2 + y^2 + z^2}$ over unit sphere

I'm studying triple integrals (physics major), and I'm having trouble solving this little beast: $$ \iiint_V \frac{1}{\sqrt{2 + x^2 +y^2 +z^2}} \,dx\,dy\,dz$$ where V is $$x^2+y^2+z^2=1$$ Of ...
1
vote
0answers
27 views

How to estimate norms involving $|a-b|$?

I know the title isn't the best one. Here's my problem: Whenever I'm given functionals such as: $$\phi: \mathcal{C}^1([0,1]) \ni f \rightarrow f(\frac{1}{3}) - f'(\frac{2}{3}) \in \mathbb{R}, \ \ ...
8
votes
2answers
121 views

How can I show that $ \int_0^\pi \frac{x\,dx}{1+\cos^2(x)} = \frac{\pi^2}{2\sqrt{2}} $

Show that $$ \int_0^\pi \frac{x\,dx}{1+\cos^2(x)} = \frac{\pi^2}{2\sqrt{2}} $$ I tried using change of variable $x = \pi-y$ and then ended up with integral $\int_0^\pi \frac{1}{1+\cos^2(y)}dy$ which ...
1
vote
0answers
55 views

Prove the result without using complex analysis [on hold]

Prove that for $k > -1 ,n>1 $ and $n-k>1$: $$\int_{0}^{\infty} \dfrac{x^k-1}{x^n-1}dx=\dfrac{\pi}{n}[\cot \dfrac{\pi}{n}-\cot{(k+1) \dfrac{\pi}{n}}]$$
2
votes
1answer
32 views

Norm of a linear functional, sup, definite integral

I'm trying to find the norm of this mapping: $$\phi: C^1([0,1]) \ni f \rightarrow \int_0^{1/2} f(t)dt + f'(\frac{2}{3}) \in \mathbb{R}$$ with $||f|| = \sup_{t \in [0,1]}|f(t)| + \sup_{t \in ...
4
votes
1answer
20 views

Integral with absolute value of the derivative

I'm trying to estimate this integral $\int_0^1 t |p'(t)|dt$ using this value $\int_0^1 |p(t)|dt$; here $p $ is a real polynomial. This means, I am looking for an $M>0$ such that $$\int_0^1 |t ...
-3
votes
1answer
45 views

$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \,dy$

Evaluate $$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \, dy$$ Please give some hints how to proceed.
2
votes
0answers
35 views

$\int_0^\infty\int_0^\pi\frac{k^2(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}})\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$

$$\int_0^\infty\int_0^\pi\frac{k^2\left(e^{-it\sqrt{k^2+m^2}}-e^{it\sqrt{k^2+m^2}}\right)\sin(\theta)}{e^{-ikx\cos{\theta}}\sqrt{k^2+m^2}}d\theta dk$$ I saw this Integral at Quora, and I have not ...
3
votes
2answers
81 views

How to show that $ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \ln\left(\tan(x)+\tan\left(\frac{\pi}{6}\right)\right)\tan(x)\space dx=\frac{\zeta(2)}{6} $

I was trying to prove the well known result: $$ \sum_{k=1}^\infty \frac{1}{\binom{2k}kk^2}=\frac{\zeta(2)}{3} $$ and it came down to prove the following equation: $$ ...
0
votes
1answer
37 views

Relation between $\tan^{-1}(x)$ and $\cot^{-1} (x)$

Suppose we've got $$I_1=\int_{-1}^{1} \tan^{-1}(x) + \tan^{-1} \left(\frac{1}{x}\right)$$ and $$ I_2=\int_{-1}^{1} \cot^{-1}(x) + \cot^{-1}\left(\frac{1}{x}\right)$$ So how can we relate $I_1$ and ...
0
votes
1answer
25 views

Using Stoke's theorem evaluate the line integral $\int_L (y i + zj + xk) \cdot dr$ where $L$ is the intersection of the unit sphere and x+y = 0

Evaluate $$\int_L (y i + zj + xk) \cdot dr$$ where $L$ is the intersection of the unit sphere and $x+y = 0 $ traversed in the clockwise direction when viewed from $(1,1,0)$. My attempt: $∇ \times A ...
7
votes
4answers
150 views

Prove $\int_{-\pi}^{\pi}\sin \sin x \,dx=0$ without using the fact that $\sin(x)$ is odd.

Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd. Computing this in wolfram says that it is uncomputable, which leads me to believe that the only ...
5
votes
3answers
122 views

Evaluate the integral $\int_0^{\pi/2} \sin (2n x) \tan x\, dx$

Is there a elementary evaluation of the integral $ \int_0^{\pi/2} \sin (2n x) \tan x\, dx, $ where $n$ is the natural number? This number is related to the Fourier sine coefficient for $\tan (x/2)$.
2
votes
4answers
94 views

calculating a definite integral of gaussian-like form

As part of a homework question in the course "Introduction to Probability" I take, I was given the following formula: $$\int_0^\infty \exp\left(-x^2-\frac{a^2}{x^2}\right)dx = ...
1
vote
0answers
25 views

Method of stationary phase for double integrals

I am looking for a reference for the leading term in the asymptotics of a double integral over a finite rectangle R of $K(x,y)\exp(i t h(x,y))$ as $t$ goes to infinity in the following situation: the ...
0
votes
0answers
22 views

Area of the surface $S=\{(x,y,z)\mid z^2=x^2+y^2,0\leq z\leq\sqrt{x}\}$

I want to compute the area of the surface $S=\{(x,y,z)\mid z^2=x^2+y^2,0\leq z\leq\sqrt{x}\}$. Is the following attempt correct? I think a parametrization of the surface $S$ can be as follows: If ...
0
votes
0answers
45 views

2-D definite Integral [on hold]

$$I=\int_\tau^\infty \int_0^x \frac{\exp\left(-\frac xc\right)}{c(1+Py)} Q\left(\sqrt{a\frac xc},\sqrt{b \frac yd}\right) \, \mathrm{d}y \, \mathrm{d}x$$ a, b , c, d, and P are positive ...
0
votes
0answers
33 views

Is the path integral the most general representation of the inverse of the Gradient operator?

Is the path integral the most general representation of the inverse of the Gradient operator? \begin{align} \boldsymbol{\nabla} \int_{\boldsymbol{x_0}}^{\boldsymbol{x}} \boldsymbol{F} \cdot ...
1
vote
2answers
38 views

Does there exist a green's function that does not have translation symmetry?

I noticed that most Green's functions I have used take on the following functional form $G(x_1,x_2)=G(|x_1-x_2|)$. I assume these subsets of Green's functions are translationally invariant? Correct me ...
0
votes
1answer
49 views

Is this piecewise function Riemann Integrable?

Is $f(x)$ where, $ f(x) = \left\{ \begin{array}{lr} \frac{1}{x^2} & : x <0\\ x & : x \geq0 \end{array} \right. $, Riemann Integrable over $[-1,1]$? ...