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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2
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0answers
10 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
1
vote
1answer
34 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
0
votes
0answers
18 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
6
votes
2answers
70 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
0
votes
2answers
34 views

Finding the volume of revolution using the method of shells

I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about $x$-axis using the shell method. I applied the method and I got $15864/5$ ...
0
votes
0answers
14 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...
0
votes
0answers
7 views

Another mutivariable integral over a simplex

Let $p$ and $q$ be two positive integers and let $\beta \neq 1/2$ be a real number. Then let $B > A > 0$. With the help of Mathematica, ie by doing elementary integrations and by consecutively ...
4
votes
2answers
46 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
3
votes
1answer
49 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
0
votes
1answer
5 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
1
vote
2answers
42 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
1
vote
0answers
22 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
1
vote
0answers
273 views

Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$

I was fiddling with some maths and came up with an interesting relationship: $$ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$$ ...
9
votes
2answers
99 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
-1
votes
5answers
57 views

Does $\int_0^{\infty}\frac{x\hspace{1mm}dx}{x^3+1}$ converge? [on hold]

Does $\int_0^{\infty}\dfrac{x\hspace{1mm}dx}{x^3+1}$ converge? Can some explain how to approach this problem? All ideas are appreciated
0
votes
1answer
21 views

what is the order of integration for : integral of x dx * integral of y dy

I'm still trying to get my head around he basics of this stuff so please use simple language in your answer ! $$ \int dx \int f(x,y) dy$$ the first integral limits are from 0 to 1 for dx and the ...
-1
votes
2answers
22 views

Integral Differentiation over constants [on hold]

Let $f(x)$ be integrable on $x\in[0,X]$ and $a,b>0$ constants. I would like to get the derivative of $$I(x)=a\int^x_0{(b-X-f(x))dx}$$ with respect to $b$, i.e. $\dfrac{\partial I(x)}{\partial b}$. ...
0
votes
0answers
27 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
31 views

How to solve the following integral [on hold]

Do you have any idea how to solve the following integral: $$\int\limits_0^a {{e^{\large \left(- \frac{{by}}{{c - dy}} - ey\right)}}dy}$$ where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you ...
12
votes
5answers
131 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
0
votes
0answers
35 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
0
votes
0answers
8 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...
3
votes
2answers
88 views

How to show that $\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}$?

From numerical evidence it appears that whenever the integral converges, $$J_a :=\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}.$$ For $a \in \mathbb{N}$, I was able to prove this using ...
1
vote
0answers
17 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
1
vote
2answers
35 views

Integral of pdf

I need to find the integral for this pdf but I don't know if I need to, or can, take the integral of two variables at the same time. $$ f(x;\theta)=\frac{x}{\theta^2} e^{-x^2/(2\theta^2)} ,\quad ...
2
votes
1answer
47 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
-2
votes
0answers
17 views

Solve definite integral using complex-variable technique. (Engineering mathematics class) [on hold]

Solve the definite integral using the complex-variable technique.
0
votes
3answers
52 views

how to integrate $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $? [on hold]

what to find $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $ ? Could you give me a hint? Thanks!
-4
votes
1answer
34 views

integrate by parts: $\int \cosh^2(x)dx$ please show solution step by step [on hold]

Integrate by parts: $$\int \cosh^2(x)dx$$ Please show the solution step by step. I actually somehow found my self in a loop solving the integral: = cosh(x) sinh(x) - int (sinh(x) (-sinh(x)) (x) = ...
1
vote
2answers
35 views

Integration with square root in denominator

I am honestly embarrassed to ask this because i feel like i should know how to do this but: $ \int \frac{x}{\sqrt{2x-1}}dx $ Try to use u-substitution please
0
votes
2answers
28 views

Gamma function in $C^{2}$

How can I show that for $x>0$, the Gamma function is at least $C^{2}$? The Gamma function is defined as $$\displaystyle \int^\infty_0 e^{-t}t^{x-1}\ dt$$ For which $x$ is the integrand integrable?
1
vote
2answers
83 views

How to integrate $\int \frac{\sqrt{x}}{x+1}dx$?

How to integrate $$\int \frac{\sqrt{x}}{x+1}dx$$ Can I substitute $x+1$ with $u$?
0
votes
1answer
19 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
votes
1answer
17 views

Fubini's theorem, interchanging order of integration

My question is, imagine I want to compute the following integral: $$\int_A \int_B f(x,y)dxdy$$ and I decide to start from $x$ and get $$\int_A \int_B f(x,y) dxdy <\infty.$$ On the contrary if I ...
2
votes
0answers
28 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
0
votes
0answers
11 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
0
votes
2answers
29 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
5
votes
0answers
38 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
3
votes
1answer
44 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}^{2 \pi} f(z) \, dz. $$ Isolated ...
1
vote
2answers
36 views

Calculus: volume of revolution about a line other than the $x$-axis.

Find the volume of the solid of revolution obtained by rotating the region bounded by $f(x) = x^3 + 1$, $g(x) = x^2$ and $0 ≤ x ≤ 1$ about the line $y = 3$. I know the gist of the problem, but ...
-1
votes
2answers
27 views

Calculate line integral

For $f(x,y) = 2x + y + 10$, calculate the line integral $$ \int_{L}{f(x,y)dL} $$ where $L$ is the straight line between $(1,4)$ and $(5,1)$ in the $xy$-plane.
6
votes
3answers
85 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
0
votes
1answer
41 views

Integration by Substitution

Question: Use the substitution $u=\tan x$ to find $\displaystyle \int_{0}^{\large \frac{\pi}{4}}\left(\tan^{n+2}x + \tan^{n}x\right)dx$. Using the above result, find the exact value of ...
2
votes
0answers
11 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
1
vote
1answer
21 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
0
votes
2answers
16 views

Integration separation of variable

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, t minutes later, the volume of liquid in the tank is V cm3 . The liquid is flowing into the tank at a constant ...
2
votes
1answer
57 views

Closed-form of $\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx$

We know that $$\int_0^1 \int_0^1 x^y\,dy\,dx = \ln 2.$$ Do we know a closed-form of $$\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx\,?$$ As a start we know that $$\int_0^1 x^{(y^z)}\,dz = ...
0
votes
2answers
30 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
4
votes
1answer
57 views

Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$

I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work. The answer is ...
2
votes
2answers
58 views

Evaluate $\int\sec^4(u) \operatorname d \!u$

Evaluate $$\int\sec^4(u) \operatorname d \!u$$ I don't know what to substitute: I've tried $1+\tan(u)$ and integration by parts. I know the general formula for $\sec^n(u)$, but I want to be able to ...