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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6
votes
1answer
86 views

Integration by parts: How to choose the constant which make calculations easier?

The formula of integration by parts is: $$\int u(x)v(x) dx = u(x)V(x) - \int u'(x)V(x) dx$$ Which can be re-written as: $$\int u(x)v(x) dx = u(x)[V(x)+C] - \int u'(x)[V(x)+C] dx$$ where C is a ...
6
votes
1answer
125 views

What is the integral of a function from $\infty$ to $\infty$?

I apologize in advance for my bad English, I am italian. I took a Calculus exam today, and one of the exercises was: $$ \lim_{x\to\infty} \int_x^{x+1/x}e^{t^2}dt $$ I answered $0$ even though I knew ...
3
votes
0answers
60 views

Transform an improper integral with $\sin,\cos,\ln$ into an equal integral over $[0,2\pi]$

As there are many properties of integrals and methods of integration often some seem to escape being "readily seen how to...". This is one of the many. The integral in question is \begin{align} ...
7
votes
1answer
187 views

Computing a double gamma-digamma-trigamma series

What are your thoughts on this series? $$\sum _{k=1}^{\infty } \sum _{n=1}^{\infty } \frac{\Gamma (k)^2 \Gamma (n) }{\Gamma (2 k+n)}((\psi ^{(0)}(n)-\psi ^{(0)}(2 k+n)) (\psi ^{(0)}(k)-\psi ^{(0)}(2 ...
2
votes
2answers
70 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
6
votes
2answers
87 views

Evaluate $\int_{-1}^{1}\int_{x}^{2x-1}dydx$

$$\int_{-1}^{1}\int_{x}^{2x-1}dydx$$ $$ My attempt: $$I_1:=\int_{-1}^{1}\int_{x}^{2x-1}dydx$$ $$=\int_{-1}^{1}\bigg[\int_{x}^{2x-1}dy\bigg]dx$$ ...
1
vote
1answer
47 views

find the integral using Integration by partial Fractions

Here is my work for this problem...just wanted a check over and see if i missed anything Original Problem: $\int$ $\frac6{x^3-3x^2}$ F 6/x^3-3x^2= F 6/x^2(x-3) 6/x^2(x-3)= Ax+B/x^2+C/(x-3) ...
6
votes
4answers
154 views

calculate $\int_{0}^{\pi} \frac{dx}{a+\sin^2(x)} $using complex analysis

where $a>1$ calculate $$\int_{0}^{\pi} \dfrac{dx}{a+\sin^2(x)}$$ I tried to use the regular $z=e^{ix}$ in $|z|=1$ contour. ($2\sin(x) = z-\frac1z)$, but it turned out not to work well because ...
-1
votes
1answer
37 views

Distribution of stochastic integral w.r. to brownian motion

Let $B=(B_t)_{t \geq 0}$ be a standard brownian motion, $T > 0$ and $f : [0,T] \rightarrow \mathbb{R}$ a continuous function. I want to determine the distribution of the following integral: ...
1
vote
3answers
76 views

In Riemann-Stieltjes Integration why do we take $\alpha$ to be monotonically increasing?

In Riemann Stieltjes integral for the function $f$ with respect to $\alpha$ why do we always take the function $\alpha$ to be monotonically increasing? Cant we take the function $\alpha$ to be ...
1
vote
1answer
39 views

Improper integral existence exercise

$$\int _0^1\frac{\ln (1+\sqrt{x})}{\sin(x)} \, dx\:$$ So the singularity point is at $0$, so we`ll use this test: $$\lim _{x\to 0}\frac{\frac{\ln(1+\sqrt{x})}{\sin(x)}}{\frac{1}{\sqrt{x}}}=\lim_{x\to ...
2
votes
1answer
70 views

how to solve $\int_{-\infty}^\infty e^{-x^2-x{\tau}} \cdot x\ dx$? [duplicate]

I took ${-x^2-x{\tau}} = k $ on differentiating it $dk=(-2x-\tau)dx$ but this substitution doesn't work how can i proceed further .
1
vote
1answer
63 views

Averaging for nonlinear systems

I am trying to figure out how the following result has been obtained. Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system: $$ \dot{ \hat{x} }(t) = k a \sin ( \omega ...
4
votes
1answer
53 views

Evaluate $\oint_{C}(y-\sin x)dx+\cos x dy$

$$\oint_{C}(y-\sin x)dx+\cos x dy$$ triangle:$$C=\{y=0,x=2\pi,\pi y=2x\}$$ My attempt: Using Green's theorem $$\oint_{C}\underbrace{(y-\sin x)}_{P}dx+\underbrace{\cos x}_{Q} ...
1
vote
0answers
21 views

Proof of differentialbility in mean square calculus?

let $x_t$ be a mean squared Riemann integrable over $[a, t]$ for every $t\in[a,b]$. Then $y_t=\int\limits_a^t x_\tau d\tau\ $ is mean squared continuous on $[a, b]$. Furthermore, if $x_t $ is mean ...
0
votes
1answer
23 views

Integrals involving hypergeometric functions

Can anyone please help me in finding the integral $\int\limits_{0}^{1/2}\frac{\left|t-\frac{1}{6}\right|}{((1-t)a+tb)^{2}}t^{s}\mathrm{d}t$ where $0<a<b$ and $0<s\leq1$? Thanks in advance.
0
votes
2answers
41 views

Integration with switch of differential

I am looking to solve integral with form $$\int_a^b av\, dt$$ where $a=\frac{dv}{dt}$ is acceleration and $v=\frac{ds}{dt}$ is speed. Is following solution correct? $$\int_a^b av\,dt=\int_a^b ...
1
vote
1answer
49 views

Evaluate $\iiint e^{-x^2-2y^2-3z^2}dV$

Evaluate the triple integral in $\mathbb{R}^3$ of $$\iiint e^{-x^2-2y^2-3z^2}dV$$ The hint is: $$\int e^{-x^2}dx = \sqrt \pi$$ I saw the same exercise here :$\iiint e^{-x^2-2y^2-3z^2}dV$ but I ...
3
votes
0answers
36 views

Can an integral be certifiably non-elementary?

If you want to convince somebody that a particular natural number is prime, you can hand them a primality certificate--a small bundle of data which can be used to efficiently generate a proof of ...
2
votes
2answers
88 views

How to find the integral $\int \frac{\sqrt{1+x^{2n}}\left(\log(1+x^{2n}) -2n \log x\right)}{x^{3n+1}}dx$?

How to evaluate the integral : $$\int \frac{\sqrt{1+x^{2n}} \, \left(\ln(1+x^{2n}) -2n \, \ln x \right) \, dx}{x^{3n+1}}$$ I have attempted an evaluation, but I am at a loss as to a useful result. ...
2
votes
3answers
118 views

Best approach to solve this integral?

I'm lost. I tried u-substitution and integration by parts, and ended up with nothing good. By parts just repeats itself over and over again. What should I do? $$\int(e^{-t}+3e^{-t}\sin(t))dt$$ The ...
4
votes
2answers
144 views

Is $\int_a^a f(x) dx$ always zero?

Is the result: $$\int_a^a f(x) \,\text{d}x = 0$$ always zero? This seems obvious at first, but what if $f(x)$ diverges at $x=a$? For example, Wolfram Alpha tells me $$\int_0^0 ...
0
votes
1answer
50 views

Partial integration (not integration by parts), how is the following result obtained?

Given we have these three equations: $\frac{\partial^2w}{\partial x^2} = D_1$ (1) $\frac{\partial^2w}{\partial y^2} = D_2$ (2) $\frac{\partial^2w}{\partial x\partial y} = 0 $ (3) Where $D_1$ and ...
0
votes
4answers
112 views

When $f(0) = 0,$ is it always true that $G(0) = 0,$ where $G$ is the antiderivative of $f$?

I have a hunch that it is, but it would be nice if somebody could confirm / disprove it for me. Thank you. Edit Is it when the constant of integration is equal to zero?
1
vote
0answers
45 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
0
votes
1answer
81 views

Evaluate$\int_{-2}^2\int_{y^2-3}^{5-y^2}dxdy$

I evaluated this integarl: $$\int_{-2}^2\int_{y^2-3}^{5-y^2}dxdy=\boxed{\frac{64}{3}}$$ Now I need to evaluate it with changing the limits, My attempt: $$=\int_{-3}^1 ...
0
votes
1answer
78 views

What happends when we multiply an integral by itself? [closed]

What happens when we multiply an integral by itself? $$ \left(\int_{z}^{q} y(x)\;\mathrm{d}x\right)^2 = \left(\int_{z}^{q} y(x)\;\mathrm{d}x\right) \left(\int_{z}^{q} y(x)\;\mathrm{d}x\right) ...
2
votes
2answers
93 views

Find a function $f(x)$ in an integral

(Related question here). Is there a way to calculate the function $f(x)$ in this integral in terms of $x$ without using $a,b,c$: $$\int_{a}^{b} f(x)dx=c$$ Two examples $\rightarrow$ how do find ...
1
vote
1answer
147 views

Find Limit Using Lebesgue Dominated Convergence

I'm trying to find the following limits using Dominated Convergence Theorem, but can't seem to find a dominating function. Any guidance would be greatly appreciated! $\lim\limits ...
-1
votes
4answers
98 views

Calculate function: $\int_{a}^{b} \left(f{(x)}\right)dx=c$

Is there a way to find the function $f{(x)}$ for a given value of $a,b,c$? $$\int_{a}^{b} \left(f{(x)}\right)dx=c$$ For example: $a=0,b=1,c=\frac{1}{3}$ we get: $$\int_{0}^{1} ...
4
votes
0answers
83 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
1
vote
2answers
54 views

How to find the integral $F = \int_a^b \dfrac{e^{-x}}{\sqrt{x-a}}dx$

Is there a solution in closed form for the following integral? \begin{equation} F = \int_a^b \dfrac{e^{-x}}{\sqrt{x-a}}\,dx \end{equation}
1
vote
3answers
69 views

How to find the original function from a definite integral.

I have that $\int_{0}^{x} f(x) \,dx = 2x,$ and I would like to find $f(x)$. I am not even sure how to begin. I would appreciate any help!
-1
votes
1answer
76 views

Evaluate $\int_{-2}^2\int_{y^2-3}^{5-y^2}dydx$

$$\int_{-2}^2\int_{y^2-3}^{5-y^2}dydx$$ My problem is that I don't see the area that I should calculete, it looks weird Attempt: $$\int_{-2}^2 \bigg( (5-y^2)-(y^2-3)\bigg) \color{red}{dx}$$ ...
0
votes
1answer
51 views

Prove that the value of an integral is negative for arbitrary distribution

Consider the integral given by $V(\Lambda) = \int_0^1 F(x) [1 - \Lambda +\Lambda F(x)] [x f(x) - 1 + F(x)] - [1-F(x)][1 + \Lambda F(x)][F(x) + f(x) x]\; dx $, where $F$ is the cdf of some ...
1
vote
0answers
54 views

How to evaluate this integral when $\lambda < \frac{3}{4}$

$$\frac{2^{2 \lambda -1} \lambda ^2}{\Gamma \left(\frac{3}{2}-\lambda \right)^2}\times\int_0^\infty \left(\frac{q}{z}\right)^{2-2 \lambda } K_{\lambda -\frac{3}{2}}(q z) \left[\left((1-4 \lambda ) ...
-4
votes
2answers
75 views

Find the value of the given integral

Find the value of the integral - $$\int \cfrac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x}dx $$ EDIT : This is what I've tried $$\int\cfrac{\cos x (\cos^2 x + \cos^4 x)}{\sin^2 x + \sin^4 x} dx \\ ...
3
votes
1answer
81 views

How to find $F = \int_a^\infty \frac{e^{-x}}{\sqrt{x-a}}\,dx$

Is there an analytic expression for the following integral? \begin{equation} F = \int_a^\infty \dfrac{e^{-x}}{\sqrt{x-a}}\,dx \end{equation}
2
votes
1answer
71 views

Evaluate $\iint_{R}(x+y)^2dxdy$ in $0\leq r\leq 1 \,\, ,\frac{\pi}{3}\leq \theta\leq\frac{2\pi}{3}$

$$\iint_{R}(x+y)^2dxdy$$ $$0\leq r\leq 1 \,\, ,\frac{\pi}{3}\leq \theta\leq\frac{2\pi}{3}$$ My attempt number 1: $$=\iint_{R}(x^2+2xy+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
0
votes
1answer
38 views

integral, domain bounded by planes

I have to determine the volume of the domain which is bounded by the plane $x=0, y=0, z=-2, z=4-x-y$. I have an integral$$\iiint 1 \,dx\,dy\,dz$$ but I don't know the limits. Could you explain me how ...
0
votes
0answers
72 views

Integration of a given Integral

Given the integral $$ \hat{\alpha}({r_{0}})=2\int^{\infty}_{{r_{0}}}\frac{dr}{r \sqrt{1-\frac{2M}{r}} \sqrt{\left(\frac{r}{{r_{0}}}\right)^{2}\left(1-\frac{2M}{{r_{0}}}\right) ...
1
vote
2answers
76 views

Looking for a function that fits a certain criteria

I am looking for a function that fits this description: $$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$ or $$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$ For all values of $n$, with this function i am looking to ...
0
votes
1answer
82 views

Integrate $\int \frac{x}{x^4-a^4}dx$

Integrate $\displaystyle \int \frac{x}{x^4-a^4}dx$. My approach: $$\begin{align}\displaystyle\int \frac{x}{x^4-a^4}dx&=\int\frac{x}{(x^2)^2-a^4}\\ ...
0
votes
1answer
58 views

Integral of divergence equal to divergence of integral?

Just as the heading reads...is the integral of the divergence of a vector field equal to the divergence of the integral of a vector field? $\int\nabla\cdot\vec U dz = 0$ same as ...
2
votes
3answers
45 views

limits of integration and derivative

I have an integral that gives $$\left[\frac{d}{dx}[f(x)]\right]_a^b$$ is it possible in general to claim that this is equal to $$\frac{d}{dx}\left(\left[f(x)\right]_a^b\right)\text{ ?}$$ If not in ...
2
votes
2answers
43 views

Finding the area between : $\frac{\pi}{3}\leq \theta \leq \frac{2 \pi}{3}, 0 \leq r \leq 1$ using polar coordinates

I need to find the area between : $\frac{\pi}{3}\leq \theta \leq \frac{2 \pi}{3}, 0 \leq r \leq 1$ using polar coordinates $x=r\cos \theta , \, y=r\sin \theta$ $$x^2+y^2=r \Longrightarrow r=1$$ ...
0
votes
1answer
30 views

Finding the area between : $\frac{\pi}{3}\leq \theta \leq \frac{2 \pi}{3}, 0 \leq r \leq 1$

$$\iint dx \,dy $$ I need to find the area between : $\frac{\pi}{3}\leq \theta \leq \frac{2 \pi}{3}, 0 \leq r \leq 1$ My attempt: Using Fubini's theorem Using double integral: ...
0
votes
1answer
37 views

Finding area of trapezoid $\{(0,2),(2,2),(3,0),(-1,0)\}$ using double integral

$$\iint dx dy $$ I need to find the area between the trapeziod:$$\{(0,2),(2,2),(3,0),(-1,0)\}$$ My attempt: Using geometry formula:$S=\frac{(a+b)}{2}\cdot h=\frac{(2+4)}{2}\cdot ...
11
votes
5answers
128 views

Proving the integral series $\int _0^1\left(1-x^2\right)^n\,dx=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}$

We have the series $\left(I_n\right)_{n\ge 1\:}$ where $$I_n=\int _0^1\left(1-x^2\right)^n\,dx.$$ Prove that $$I_n=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}.$$ I tried to ...
-2
votes
3answers
72 views

How $\cos ^2\left(x\right)\sin ^2\left(x\right)=\frac{1-\cos \left(4x\right)}{8}$ [closed]

$$\cos ^2\left(x\right)\sin ^2\left(x\right)=\frac{1-\cos \left(4x\right)}{8}$$ $2\cos \left(x\right)\sin \left(x\right)=sin2x$