Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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2answers
47 views

Evaluate the following Integration--

Evaluate the following Integration $$\int \frac{\cos^9 x}{\sin^3 x + \cos^3 x} \,dx$$ I tried, but this problem is very difficult to me. any help?
2
votes
2answers
108 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
10
votes
2answers
209 views

Evaluating $\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ dx$

One of the ways to compute the integral $$\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ ...
0
votes
3answers
48 views

A general method for integration of rational function.

$\int\frac {x^3}{1+x^5}$ ATTEMPT: I did the following substitution: Let $x=\frac{1}{t}.$ $dx=\frac{-1}{t^2}dt.$ substituting back: $I=\int\frac{-1}{1+t^5}dt$ which doesn't seems a simpler ...
13
votes
1answer
187 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion [duplicate]

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
2
votes
0answers
26 views

$f(x) = lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on $[0,1]$

A function defined on $[0,1]$ by $f(0) = 0$ and $f(x) = \lim_{n \to \infty} \frac{(1+ \sin \frac{\pi}x)^n - 1} { (1+ \sin \frac{\pi}x)^n +1}$, $x \in (0,1]$. To show that $f$ is integrable on ...
2
votes
2answers
35 views

Solving this Integral with Bessel Functions

Any suggestions on solving this (J0,J1 Bessel function of first kind, 0th and 1st order, respectively) : $$ T = \int_0^a \int_0^\infty J_0(\lambda r) J_1(\lambda a) ...
4
votes
3answers
79 views

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$?

Why is $\int_{0}^{\pi}{1\over 1-\sin x}dx=2\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$, or to be accurate: why is $\int_{\pi\over 2}^{\pi}{1\over 1-\sin x}dx=\int_{0}^{\pi\over 2}{1\over 1-\sin x}dx$? ...
0
votes
0answers
19 views

Area of region - Double integrals

Here is my task: Calculate area of region $(\frac{x}{a}+\frac{y}{b})^{5}=\frac{x^{2}y^{2}}{c^{4}}$,$a,b,c>0$. Solution is $A=\frac{a^{5}b^{5}}{1260c^{8}}$ Any idea how to solve this?
0
votes
1answer
50 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
1
vote
2answers
126 views

Solving a given complex integral

I am trying to solve a problem that involves solving the integral $$\int\frac{1}{\sqrt{y^2 + a^2}} \left(\frac{\sqrt{y^2 + a^2}}{k} - 1\right)^pdy$$ Where $$p=1-\frac{1}{1+n}, n>1$$$, $n$ is an ...
11
votes
1answer
134 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
0
votes
0answers
10 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
0
votes
2answers
32 views

(definite integral) area between two trig functions

I'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever ...
5
votes
1answer
107 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
6
votes
4answers
187 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
0
votes
6answers
104 views

What happens to an integral if the 'dt' disapears? (Integral with no dt)

I have been struggling with convolution yet have come up with my own method that involves cancelling the dt from the integral. I want to ask - If there is a term that is - ''Integral'' with limits ...
2
votes
0answers
45 views

Gamma function still hard for me

During my study I find a form for gamma function it was $\Gamma (x) = \lim_{n\to\infty} \frac{n! n^{x-1}}{x(x-1)(x-2)........(x+n-1)}$ And then by simplify this limit I get $$\lim_{n\to\infty} ...
6
votes
1answer
120 views

A difficult integral: $\int_0^{+\infty} e^{ - x}\left(\frac{1}{x( e^{ - x} - 1 )} + \frac1{x^2} + \frac1{2x} \right) \, dx$.

Could you help me calculate the integral? $$\int_0^{+\infty} e^{ - x} \left( \frac{1}{x( e^{ - x} - 1)} + \frac{1}{x^2} + \frac{1}{2x} \right) \, dx .$$
0
votes
1answer
22 views

How would you integrate this homogeneous equation?

I am solving a homogeneous equation $\frac{dy}{dx}= \frac{x^2+xy+y^2}{x^2}$ and have come to this step and I'm stuck now with the integration. I could really use some helpful hints to help me $$ ...
0
votes
0answers
22 views

Integral gaussian hypergeometric function

How can we define integral with interval $[b,\infty)$ $$ \begin{align} C(b,\alpha) & = \int_b^\infty \frac{1}{1+w^{\alpha/2}}\,\mathrm{d}w \\[8pt] & = 2\pi/\alpha \csc(2\pi/\alpha)-b_2 F_1 ...
7
votes
1answer
92 views

A tough integral:$\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$

I would like to prove the convergence of $$I=\int_0^{+\infty}\left( \frac1{\log(x+1)-\log x}-x-\frac12\right)^2 dx$$ then obtain a closed form of $I$. Convergence is ensured by the fact that $x ...
0
votes
1answer
48 views

Integrate the function by substitution method.

$$\int \frac1{ \cos(x-a)\cos(x-b)} \, \mathrm{d}x$$ Can someone help me to integrate this function by method of substitution.I am not able to start it for possibilities are not coming in my mind. ...
5
votes
2answers
202 views

Evaluating the indefinite integral $\int\log\!\left(x+\sqrt{x^2-1}\right)\!dx$

I came across the following integral, and I don't know how to solve it. $$ \int\log\left(x+\sqrt{x^2-1}\right)dx $$ I tried the "obvious" substitution of $x=\sec\theta$, which gives you: $$ ...
1
vote
3answers
53 views

How can i prove that this integral is convergent/divergent

This is my equation: $$\int_0^{\pi/4} \frac{dx}{x\sin2x}$$ I wish to prove that it's convergent or divergent, by $P$ test and/or comparison test, but it does not seem to be applicable... Is it ...
16
votes
1answer
261 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
0
votes
1answer
17 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
6
votes
0answers
51 views

The quadratic and cubic versions of a tough intregral

In this post, Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$, it's proved that $$I_1=\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log ...
2
votes
3answers
39 views

Did I integrate correctly? Integration using algebraic substitution.

Integrate with respect to $x$ $$\int3{\sec^2(3x)\tan(3x)dx}$$ There's 2 ways of doing this according to the book, I just wish to know if I did both ways correctly...please correct me on where I ...
0
votes
2answers
71 views

Integral of $x/(2x-1)$

I'm not sure how to do this, I'm also new to math.stackexchange so please excuse any novice mistakes. So anyways, here is a question I have on a summer assignment for Calculus BC (this is review from ...
0
votes
2answers
23 views

Integration of probabilities

I am trying to find a way to properly formulate a specific integral. Given two probabilities: $$ f(x(0))=\begin{cases} 1/2, & -1\le x(0) \le 1\\ 0, & \text{else} \end{cases} $$ and $$ ...
1
vote
1answer
41 views

$\int \frac{q}{\sqrt[3]{k^3-q^3+\frac{3 }{8}q^2}} \, dq$ where $k = 0.14$

$$\int \frac{q}{\sqrt[3]{k^3-q^3+\frac{3 }{8}q^2}} \, dq$$ I have tried solve this by substituting $k^3-q^3+\frac{3}{8}q^2 = u$ but this wont work because of the higher order polynomials it has. ...
7
votes
4answers
236 views

Integral involving a trig. term

I came across the following integral. $$ \int\frac{dx}{1+\sin x} $$ I have no idea how to solve it! I went for the obvious substitution of $u=1+\sin x$, but then you get an annoying $\cos x$ kicking ...
3
votes
1answer
46 views

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: ...
0
votes
0answers
33 views

Which way to do this integral?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
3
votes
1answer
60 views

Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to $1$ and has $g(1)=0$, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=\ln(1/x)$. I believe ...
3
votes
0answers
34 views

Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
9
votes
2answers
1k views

Integration of powers of the $\sin x$

I have to evalute $$\int_0^{\frac{\pi}{2}}(\sin x)^z\ dx.$$ I put this integral in Wolfram Alpha, and the result is ...
1
vote
0answers
23 views

Analytic or numeric integration of Singular integral based on Bessel K0 and K1

thank you for reading this ! I need the following integral to be integrated from -1 to 1. It appears to have a singularity in -2/3. In string, Generated: ...
0
votes
1answer
30 views

Interchange Order of Integrals

Can someone explain the last step in this process. Specifically, how do you get the new limits of integration? Expected Value Definition: $E[Y] = \int_0^\infty{P\{Y \ge y\} \, dy}$ Expand: $E[Y] = ...
1
vote
4answers
42 views

When can I use the natural log to help solve an integral?

Why is it okay to do this: $\int \frac{1}{x-2}dx = \ln(x-2)$ but not this: $\int \frac{1}{1-x^2}dx = \ln(1-x^2)$
1
vote
1answer
21 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
1
vote
2answers
32 views

Finding the integral of rational function of sines

$$ \int \frac{\sin x}{1+\sin x} \, \mathrm{d}x$$ How do I integrate this? I tried multiplying and dividing by $ (1- \sin x) $.
1
vote
3answers
87 views

Solve integrals using residue theorem? [on hold]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
4
votes
1answer
48 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
6
votes
2answers
50 views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

closed form for : $$\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{\mathrm{dx}}{(b^2+x^2)}$$ where $\beta$ is beta function I tried with the definition of beta and i got ...
0
votes
1answer
88 views
+100

Change of variable formula for the image of a hypercube

Let $\varphi: \mathbb{R}^n\to \mathbb{R}^n$ be an injective $C^1$ map. Let $I=[0, 1]^n$. I want to show that $$m(\varphi(I))=\int_I \left|\det D\varphi(x)\right|dx.$$ This is a special case of the ...
3
votes
2answers
66 views

Geometry of Riemann Stieltjes integration

What is the geometrical interpretation of Riemann Stieltjes Integration ? We know that for Riemaan integration $\int_{a}^b f(x)dx$ represents the area bounded by the curve $y=f(x)$& the ...
0
votes
1answer
37 views

When calculating joint probabilities using double integrals…

When calculating joint probabilities using double integrals, do we use $dx\ dy$ or $dy\ dx$ ? I thought it was the former, but then my book abruptly changes to using $dy\ dx$ without an explanation ...
0
votes
2answers
55 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...