Tagged Questions

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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0
votes
1answer
22 views

Prove reduction formula for $\cos^m (x)\sin^n (x)$

$\displaystyle\int \:\sin^m\left(x\right)\cos^n\left(x\right)dx=\frac{\sin^{m+1}x\cos^{n-1}x}{m+n}+\frac{n-1}{m+n}\int \:\sin^mx\cos^{n-2}xdx$ I have been trying to solve for over a week now can ...
4
votes
5answers
54 views

Prove $(b-a)\cdot f(\frac{a+b}{2})\le \int_{a}^{b}f(x)dx$

Let f be continuously differentiable on $[a,b]$. If $f$ is concave up, prove that $$(b-a)\cdot f\left(\frac{a+b}{2}\right)\le \int_{a}^{b}f(x)dx.$$ I know that (and have proved) $$(b-a)\cdot ...
0
votes
1answer
17 views

Evaluating line integrals in the plane

Evaluate the integral of xy(dx)+(x+y)(dy) along the curve y=x^2 from (-1, 1) to (2, 4). I tried finding dx and dy and substituting that into the original integral, along with y=x^2. But I don't get ...
0
votes
0answers
10 views

A doubt in Kreyszig's Functional analysis

As an application of the Uniform boundedness Theorem, it is proved in Kreyszig's Functional analysis that "There exist real valued continuous functions whose Fourier series diverge at a given point". ...
-2
votes
1answer
31 views

solve the integral$\int\frac{x}{(\csc x)-x-x^2}dx$

$\int\dfrac{x}{(\csc x)-x-x^2}dx$ Its first time I solve same this integration I'm not sure what I can do. I had though the let $u=x~dx$ and $dv=\dfrac{dx}{(\csc x)-x-x^2}$ I can't solved it.
3
votes
1answer
25 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
1
vote
0answers
16 views

Magnus series expansion

In the theory of the Magnus series expansion, it can be found that $$ \Omega(t) = \int_0^t A(\tau)d\tau - \frac{1}{2}\int_0^t \left[ \int_0^\tau A(\sigma)d\sigma, A(\tau) \right]d\tau + ...
0
votes
0answers
17 views

Sobolev space on closed surfaces

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical ...
6
votes
2answers
60 views

How does $\int_0^\infty e^{-t^4}dt = \Gamma (\frac{5}{4}) ?$

My text book claims that $$\int_0^\infty e^{-t^4}dt = \Gamma \left(\frac{5}{4}\right).$$ I fail to see this. By the definition of the gamma function we have $$\Gamma (z) = \int_0^\infty ...
-1
votes
1answer
27 views

Path integral in the complex plane

I am trying to solve this question, but I'm unsure how to parameterize it. Can you explain how I get this to an integrable form?
3
votes
0answers
38 views

Integral of cosine over a quadratic

I need help with the following integral: $$ \int_{-\pi}^{\pi} \frac{cos(ax)}{1-bx^2} dx $$ The constants $a$ and $b$ are both real and positive. Any help will be appreciated :)
1
vote
2answers
21 views

Can the integral get small outside a set with finite measure?

Let $(X, \mathcal{A}, \mu )$ be a measure space and let $ f : X \rightarrow \mathbb{\overline{R}}$ integrable. I just proved the fact that for every $\epsilon > 0$ we find $\delta > 0$ so that ...
8
votes
1answer
76 views

Ways to prove $\displaystyle \int_0^\pi dx \dfrac{\sin^2(n x)}{\sin^2 x} = n\pi$

In how many ways can we prove the following theorem? $$I(n):= \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$$ Here $n$ is a nonnegative integer. The proof I found is by considering ...
6
votes
2answers
109 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
3
votes
1answer
54 views

How solve $\int \frac{dx}{(x^2-x)^x}$

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
1
vote
1answer
15 views

If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: ...
3
votes
2answers
46 views

Improper Integral of $\int\frac{dx}{(2x-1)^3}$

Improper Integral of $$\int_{-\infty}^0\frac{dx}{(2x-1)^3}$$ from Anton Calculus 8th Edition, page 576, question 9. Answer is $-\frac{1}{4}$ but I'm finding $-1$ The integral, substituting ...
1
vote
3answers
79 views

Evaluating $\displaystyle\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$

Evaluate the following limit: $$\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$ Any advice on how to tackle this problem ?
5
votes
3answers
71 views

Why is the area under $\frac1{\sqrt{x}}$ finite and the area under $\frac1x$ infinite?

If this integral is calculated the normal way, $$\int_0^1 \frac1{\sqrt{x}} dx = 2\sqrt{1}-2\sqrt{0}=2$$ However, the graph of $\dfrac1{\sqrt{x}}\to \infty$ as $x\to 0$, so the area under the graph ...
1
vote
0answers
11 views

triple integral reduction

I have a triple integral of this kind $$\int_0^t{dx f(x)\int_{t-x}^{\infty}{dy g(y)\int_{t-x-y}^{t+\Delta t-x-y}{dz\delta(z)h(x,y,z)}}}$$ where $\delta$ is the Dirac Delta function and the other ...
2
votes
1answer
25 views

Help find error bound of trapezoidal quadrature

I'm having trouble finding the error bound of this function. My professor says it's "trivial" and thus, he refused to offer me any help beyond a simple hint I kind of knew anyway. I am given this ...
2
votes
1answer
38 views

Using $\log$ and $\ln$ in Integration [duplicate]

I found in some integral equations where they use $\log(n)$ and in some other with $\ln(n)$. Suppose $$ \int_{n_0}^{\large\frac{n_0}{2}} \frac{1}{n}dn $$ Which formula should I use ? $$ \log(n)\ ...
2
votes
1answer
16 views

Changing the order of integration of triple integral

We have a solid in a first octant, $x,y,z \ge 0$, which is bounded by the coordinate planes, the plane $x + y = 2$ and the parabolic cylinder $z + {y^2} = 1$. The task is to change the order of ...
2
votes
4answers
91 views

Evaluating $\displaystyle4\int \frac{\tan^2x\:\sec\:x}{\sec\:x\:+1}dx$

I was solving following integral $$\int \frac{\sqrt{x^2+4}}{\frac{x}{2}+1}dx$$ I think I need do a trigonometric substitution but I eventually end up with $$4\int ...
1
vote
1answer
42 views

Demostrate $\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$ [on hold]

Demonstrate: $$\int \frac{dx}{(a\sin x+b\cos x)^{n}} = \frac{A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+c \int \frac{dx}{(a\sin x+b\cos x)^{n-2}}$$ $A,B$ are undetermined coefficients
4
votes
2answers
56 views

how to compute this definite integral if possible?

how to solve this integral? $$\int_0^a\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac{3}{2}}$$ my attempt $$ \int_0^a\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= ...
0
votes
0answers
31 views

Marginalizing product of multivariate normal distributions

How should I marginalize $F_{i}$ from the following probability distribution $$p(y_{i}|F_{i},\alpha, \Lambda, \Phi, \Sigma) = N(\alpha + \Lambda F_{i}, \Sigma)$$ in order to obtain ...
2
votes
1answer
27 views

One measurable $\lim$ and one theorem

How we can prove following theorem? Let $f_n \ge 0 $ be measurable, $\lim f_n = f $ and $f_n \le f$ for each $n$. Show that $$\int f(x)dx=\lim_n \int f_n(x)dx $$ Any idea would be highly ...
1
vote
0answers
30 views

Finding the area inside the rose

Calculate the area inside the rose:$$r(\theta)=a\cos(n\theta)$$ where $n$ is a positive integer and $a$ a positive constant. Why I'm making a mistake ...
2
votes
3answers
97 views

Where did I go wrong in this integration $\int\frac{\ln(1-e^x)}{e^{2x}}\,dx$

Here is the closest I've come to the answer Link to Wolfram equality not giving true as output NB! I marked where I'm unsure in $\color{red}{red}$ color. And please don't get startled because of the ...
3
votes
2answers
46 views

Integration problem, math analysis

Suppose that $f$ is continuous on $[0,1]$ and $$ \int_0^1 x^kf(x)\ dx=0$$ for $k=0,1,...,n-1$, $$\ \int_0^1x^nf(x) \ dx=1$$ Prove that there exists $\xi\in(0,1)$ such that $|f(\xi)|\geq 2^n(n+1)$. So ...
0
votes
1answer
12 views

Gamma function identity used in deriving negative binomial from gamma-poisson mixture

On this wikipedia page negative binomial distribution, Negative binomial was derived as integrating out the lambda from Gamma-Poisson mixture. I tried to follow the proof step by step, but I am stuck ...
-1
votes
0answers
13 views

How can one take supremum and infinimum of a sum in rieman integral to define integrability?

We define a function is integrable if sup{L(f,p)}=inf{U(f,p)} where sup is supremum and inf is infinimum p is a partition function. But how can we take supremum of a number as L(f,p) is defined as ...
0
votes
3answers
49 views

Evaluate $\displaystyle\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$

Evaluate following integral: $$\int _{-1}^0\:\frac{\left(x+6\right)}{x^2+2x+2}\:dx$$ I tried to solve it but don't know how to do it, can anyone please help
2
votes
2answers
45 views

Prove that $\int^b_af(x)dx=\int^b_af(a+b-t)dt$

I'm having trouble with this two-part question: Prove that $$ \int^b_af(x)dx=\int^b_af(a+b-t)dt $$ Hence prove that, if $0<\beta<\frac{1}{2}\pi$, $$ ...
0
votes
1answer
36 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
5
votes
2answers
55 views

Evaluating $\int_0^{\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$

I've been told that this integral admits a closed form $$ \int_0^{\Large\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$$ But an integration by parts with $u'(x)=\cosh^2\left(x/\sqrt{2}\right)$ and ...
1
vote
0answers
12 views

Is the standard embedding of the torus the tight embedding?

Definition of tight: A mapping of a surface into $\mathbb{R}^3$ is called tight if its image, equipped with the induced metric, has minimal total absolute curvature. (A definition of this kind is ...
3
votes
0answers
82 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$$ I substituted $x=\tan\theta$ and got $\int_\frac{\pi}{4}^0 (\theta)^2\frac{\ln(2\sin^2\theta)}{\cos^2\theta} d\theta$ ...
2
votes
3answers
59 views

Getting hung up on notation for $\frac{d}{dx}e^u$ vs. $\int{e^u}du$

We know that $\frac{d}{dx}e^u = e^u \frac{du}{dx}$ <---Chain rule So, that means $\int{e^u}du=e^u+C$ To verify this, we could take the derivative of the integral and make sure we got back to ...
6
votes
0answers
97 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
0
votes
0answers
16 views

Taking partial of a function with two arguments inside the integral

I have a function $ f(y,w) $ which is jointly concave and twice differentiable. Let $ f^{(2)}(\cdot, \cdot) $ denote the partial of $ f $ with respect to its second argument i.e. $ f^{(2)}(y,w) = ...
4
votes
1answer
47 views

Evaluation of $\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} dt$

I need to get the value of the following definite integral $v\in \mathbb R^+$ $$\int_{-v/2}^{v/2} \sqrt{1-\left(\frac{t}{7}-\frac{v}{14}\right)^2} \sqrt{1-\left(\frac{t}{7}+\frac{v}{14}\right)^2} ...
0
votes
0answers
40 views

joint probability distribution function | Find constant c and determine if independent

I have the function: $f(x,y) = c(x+y)$ if $0 \leq x \leq 1$, $0 \leq y \leq 2$ and $0$ otherwise. 1) Find the constant c I had to set the double integral to be equal to 1. So I'm not sure if I ...
3
votes
1answer
50 views

$f(x)$ is a differentiable function for $x\in[a,b]$, and $f'(a)=f'(b)$, prove: there is a $\theta$ such that…

$f(x)$ is a differentiable function for $x\in[a,b]$ ($f'(x)$ may not continuously), and $f'(a)=f'(b)$, prove: there is a $\theta$ such that $$f'(\theta)=\frac{f(a)-f(\theta)}{a-\theta}$$ I think we ...
1
vote
1answer
37 views

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$ $$\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2$$ ...
1
vote
0answers
18 views

Triple integral, finding the volume between two planes and a surface in 3D

So I have tried to solve this problem, but I'm running into a problem, because the top circle (intersection of the function with z=1) when you project it onto the xy plane is smaller than the circle ...
1
vote
1answer
33 views

$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$, using complex transformation.

$$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$$ This integral can be done by using integrate by parts twice. However, I was thinking if this can be done by using some complex transformation (I haven't done ...
0
votes
2answers
18 views

Sign problem, Laplace transform of sin(at)

I have a problem in my integration by parts but I can't find it: $$L(\sin(\alpha t)) = \int_0^{\infty}\sin(\alpha t)e^{-st}dt$$ $$= -\frac{1}{\alpha}\left[\cos(\alpha ...
0
votes
0answers
23 views

Expansion of integrand before integration?

I have the following integral as part of a calculation $$\int_{-A}^{A} \int_{-A}^{A} \frac{1}{(z^2 + d^2)^3} dz dx, $$ where $A$ is a constant. I am given the condition $d \gg z$ so I am wondering if ...