Questions about the evaluation of specific definite integrals.

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8
votes
2answers
74 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
0
votes
0answers
32 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
votes
1answer
22 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
5
votes
1answer
176 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
45 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
35 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
42 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
53 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
112 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
15 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
41 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
48 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
192 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 ...
1
vote
1answer
23 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
24 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
76 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
83 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
109 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
31 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
84 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
48 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 ...
2
votes
0answers
29 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}, \ \ ...
9
votes
3answers
136 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
33 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
46 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
151 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
123 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
95 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 ...