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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3
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2answers
116 views

Evaluate $\displaystyle\int{\frac{e^{2x}} {\sqrt{1-e^x}}}\ dx$

Evaluate $$\displaystyle\int{\frac{e^{2x}} {\sqrt{1-e^x}}}\ dx.$$ I tried to solve by using integration by parts, but I couldn't find a solution. What method should I use to integrate this?
7
votes
5answers
206 views

Simple Integral $\int_0^\infty (1-x\cot^{-1} x)dx=\frac{\pi}{4}$.

Hellow I am trying to prove this result. $$ I:=\int_0^\infty (1-x\cot^{-1} x)dx=\frac{\pi}{4}. $$ The indefinite integral exists for this integral. The function $\cot^{-1} x$ is the arc-cotangent ...
1
vote
3answers
42 views

Finding Distance Function via integration

I would just like to confirm if: - If my answer is correct - Or find the source of my mistake. My answer: $-t^2e^{-t} -2te^{-t} -2e^{-t}$ Apologies for the ...
0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
7
votes
2answers
112 views

The antiderivative of $\cos^5(x)\sin^5(x)$ - is this incorrect?

I like to check my answers with wolframalpha, and this one's stubbornly coming up as false when set equal to its answer for the antiderivative, but I can't figure out where I'm going wrong. Using the ...
2
votes
1answer
70 views

Is the following integral equation true?

I am reading a script and I found the following statement: $$ \int_{-\infty}^\infty \frac{1}{\sigma \sqrt{2\pi}} \exp \left(\frac{-x^2}{2 \sigma^2 }\right) \exp(i \, x\, \xi) \,dx = ...
0
votes
1answer
101 views

Integral is equal to $0$

Let be $f \in L^1[0,1]$, then it applies $ \int_0^1 \exp(2i\pi xk)f(x n)\,dx=0$ for $n,k\in \mathbb{N}$ with $0<k<n$. Ideas: f can be extended to a function on $\mathbb{R}$ with period $1$, ...
0
votes
2answers
44 views

construct functions such that $f(x)g(x)\gt0 $ and

Does there exist real functions $f, g\in C^1[-1,1]$ such that $$\det\left(\begin{array}{cc}f &g \\ f'&g'\end{array}\right)\equiv0 \qquad \det\left(\begin{array}{cc}\int_{-1}^1f^2\,\mathrm ...
4
votes
1answer
75 views

Trying to integrate $\int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx$

Buenos Dias, Ciao, Hello! My fellow math stack users, I will try to solve this integral $$ \int_0^1 x(1-x)(2-x) e^{-(1-x)^2}\ln(1-x)\,dx $$ I did this $u=1-x$ $$ -\int_0^1 (u-1)u(u+1)e^{-u^2}\ln u \, ...
3
votes
2answers
98 views

How to calculate the improper integral $\int_0^\infty\left(\frac{1}{\sqrt{x^2+4}}-\frac{P}{x+2}\right)dx$

This is the first time I've seen a problem like this. I have no idea what to do. Detailed guidance would be of great help. For which values of P does the integral converge? ...
11
votes
1answer
198 views

Evaluating $\int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x$

I am trying to show that$$\displaystyle \int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x = \dfrac {2 - \sqrt 2} {8}$$ I have verified this numerically on Mathematica. I have ...
4
votes
2answers
96 views

Double integral: $\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy$

Could someone help me with this question? I am stuck on it. Compute the following double integral: $$\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy.$$ How to compute the integral when the inner ...
1
vote
1answer
59 views

Integrate $\int_a^b e^{- \cos(t)} dt$

I am looking for an explicit representation of $\int_a^b e^{- \cos(t)} dt$. The only way I could imagine to find the antiderivative is to expand this function in spherical harmonics or use the taylor ...
1
vote
0answers
50 views

Integration in physics and calculations with $dx$

I'm in a physic formation and we are used to play with the infinitesimal elements $dx$ of integration like a variable (for example the calculation of the pressure of a gas), because we look at small ...
2
votes
1answer
26 views

$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2)\; dt = \frac{(-1)^n}{c}, \text{ where } \sqrt{n\pi} \leq c \leq \sqrt{(n+1)\pi}. $

The following is a problem from Apostol Vol 1 Calculus from the section: Continuity. Since Differentiation hasn't been introduced yet, the objective is to solve it without direct reference to ...
1
vote
1answer
44 views

If $\int _1^{\infty }f\left(x\right)dx$ converges absolutely then $\int _1^{\infty }\sin \left(x\right)f\left(x\right)dx$ exists

I'm in need of some assistance with a homework question (I'm doing some calculus work by myself and have gotten stuck on this question): "Prove or give a counter-example of of the following ...
1
vote
2answers
653 views

Area of a weird ellipse shape.

A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ ...
1
vote
1answer
50 views

Gauss Divergence Theorem Calculation help

I am having trouble getting my head around what exactly is required in this problem. Let $S$ be an arbitrary piecewise smooth, orientable, closed surface enclosing a region $\mathbb{R}^3$. Calculate ...
1
vote
1answer
39 views

Solving a general integral (expectation of some variant of exponential distribution)

Suppose $X$ is distributed exponentially with parameter $\lambda$. Its pdf is $\lambda e^{-\lambda x}$, and the calculation of its expectation is straight forward: $\mathbb{E}(X) = \int_0^\infty ...
1
vote
1answer
40 views

Integration by Substitution question

I just wanted to check if I did this question correctly or if I made a mistake when calculating $\frac{du}{dx}$
1
vote
1answer
110 views

Calculus, volume between plane and paraboloid - check my answer

Fairly simple question, we have the paraboloid $z=a(x^2+y^2)$ and the plane $z=h$. $a,h >0$ Find the volume of the region bounded by the plane and the paraboloid. What I did: It is clear to see ...
1
vote
3answers
56 views

Comparison test for $\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$

$$\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$$ How come up with something to compare it to use the comparison test?
2
votes
1answer
41 views

Integral comparison test

$$ \int_a^b \frac{x+1}{\sqrt[4]{x^5+x^2}} \,dx $$ I want to know this integral converges or not when $(a, b) = (0, 1), (1, \infty)$. I was thinking of using the comparison test, but I can't think of ...
5
votes
3answers
206 views

Integration over four lines! Tricky one??

I don't know how to seal the deal on this one. If you just read the comments on the answer by @RecklessReckoner you will see where I am still stuck. I would appreciate if someone could ...
0
votes
1answer
51 views

Can't understand a reduction formula question

I can't even understand the question. Pls help me T_T Hope any one can help me answer this question. Homework too hard T_T Appreciate .... ^_^
0
votes
1answer
29 views

Probably a dumb question on integration(Geometry of a simple double integral of $x$

I want to integrate $\int \int_D x \;\mathrm{dA}$ With D being the area enclosed by a rectangle $(0,0),(3,0),(3,1),(0,1)$ Now I would think the integral should be setup as: $\int_0^1 \int_0^3 x dx ...
1
vote
2answers
64 views

Definite Trig Integrals: Changing Limits of Integration

$$\int_0^{\pi/4} \sec^4 \theta \tan^4 \theta\; d\theta$$ I used the substitution: let $u = \tan \theta$ ... then $du = \sec^2 \theta \; d\theta$. I know that now I have to change the limits of ...
1
vote
1answer
353 views

Double integrals of exponential functions

I need to find the double integral of $$e^{\frac{x}{y^2}}$$ bound by the $y\mbox{-axis}$, $x=y^2$, $y=1$, and $y=2$. The limits of integration were easy to find, but I am pretty confused about how to ...
1
vote
2answers
67 views

limit of integrals of increasing functions

Let $\{f_n\}$ be a non-decreasing sequence of integrable functions such that $\lim_{n \to \infty} f_n \geq 0$. If $S = \{x|\lim_{n \to \infty}f_n > 0\}$ is not a null set, show that $\lim_{n \to ...
0
votes
0answers
149 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
1
vote
0answers
45 views

Holomorphic Functional Calculus

Framework: Consider a Banach space: $$(E,\|\cdot\|)$$ Given an unbounded operator: $$T:\mathcal{D}(T)\to E\qquad\mathcal{D}(T)\subseteq E$$ together with its resolvent map: ...
0
votes
2answers
90 views

Integrate $\int \frac{\sin^2(x)}{1+\sin^2(x)}dx$

Got any ideas (what substitution should I use) to evaluate $$\int \frac{\sin^2(x)}{1+\sin^2(x)}dx~?$$
4
votes
2answers
129 views

Integrating $e^{e^{ix}}$

Evaluate $\int_0^{2\pi}e^{e^{ix}}dx$. Attempt: $e^{ix}=\cos{x}+i\sin{x}$, so we can write $$e^{e^{ix}}=e^{\cos{x}}e^{i\sin{x}}$$ and then use the same identity to get ...
1
vote
1answer
60 views

Uncountable set from Riemann integral

I have the following problem Let $f\geq 0$ and Riemann integrable on $[a,b]$. If $$\int_a^b f=0 $$ then the set of point $x\in [a,b]$ such that $f(x)=0$ is not countable. I'd like some hint ...
2
votes
2answers
124 views

Integral $\int \sqrt{x^2-3x+2}\ dx$

How to evaluate $$\int_3^{17} \sqrt{x^2-3x+2}\ dx \ ?$$ I tried Euler's substitution, that is $$\sqrt{x^2-3x+2}=x+t \Longleftrightarrow \frac{t^2-2}{-3-2t}+t=\frac{t^2+3t+2}{2t+3}\ ,$$ which I ...
0
votes
1answer
132 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
0
votes
0answers
30 views

Scalar potential, vector field and line integral

I've been trying to get my head around this topic for a project where I need to reconstruct a 3D surface given the estimated normals of it (Photometric Stereo). I just want to be sure I'm ...
6
votes
5answers
255 views

How to find $ \int_0^\infty \dfrac x{1+e^x}\ dx$

How to find $$ \int_0^\infty \dfrac x{1+e^x}\ dx=\ ...? $$ I don't know where should I start with. The correct answer from my textbook is $\frac{\pi^2}{12}$. This is my homework with 10 questions ...
2
votes
1answer
75 views

Line integral answer confirmation please :). I have moved the actual question to the first line.

My question: Am I meant to sub in something for $x$ and $y$ below? I believe I have now obtained the correct answer: $\oint_C \mathrm{F\cdot T \;ds} \;= 4xy + 4x^2 - 4xy - 4x^2 = 0 $ The ...
0
votes
0answers
43 views

Integral with Inverse error function

I have a challenging integral to solve involving the inverse error function, $\rm Erf^{-1}$, $\mathcal{I}(x,\beta)=\int\,_{x_c(\beta)}^x\,{\rm d}x\,\exp\left[\sqrt{2}\sigma\,{\rm ...
2
votes
2answers
158 views

Lebesgue Integral, Riemann Integral and Integrals of all sorts

I've heard people refer to the Riemann integral as a "teaching integral" and in a sequence of an analysis course at my school (which is rarely offered) we discuss the mysterious Lebesgue integral. I ...
1
vote
1answer
59 views

Calculating r(t) with line integrals

I have $F(x,y)$ equalling some $a \mathrm{i}+b\mathrm{j}+c\mathrm{k}$ is that all $r(t)$ is? What if all of $a,b,c$ are not in terms of $t$? Note: My $F(x,y)$ is a vector field. Or does it come ...
0
votes
0answers
54 views

Simple notation questions(2) and unit tangent vector question(1)

I have a vector field $F$, and a rectangle $C$ and some $T$ as a unit tangent vector to $C$ directed anticlockwise around $C$. How is $T$ calculated? Wouldn't it just be a straight line facing ...
3
votes
1answer
62 views

$\int\limits_0^4\int\limits_\sqrt x^2 \dfrac x{1+y^5}\ dy\ dx$

How to find $$ \int_0^4\int_\sqrt x^2 \dfrac x{1+y^5}\ dy\ dx=\ ...? $$ The correct answer from my textbook is $\frac1{10}\log33$. I can answer it if the integrand is $\dfrac x{1+y^2}$ so I can let ...
2
votes
3answers
89 views

Please, help with this integration problem

Consider the region bounded by the curves $y=e^x$, $y=e^{-x}$, and $x=1$. Use the method of cylindrical shells to find the volume of the solid obtained by rotating this region about the y-axis. I ...
1
vote
0answers
25 views

a.e. convergence of dilations of a function in L^p

Let $f\in L^p(R^d)$ ($p<\infty$) and $\delta_h f(x):=h^{d/p}f(hx)$ (the normalization is so that $\|\delta_h f\|_{p} =\|f\|_p$). Consider $(h_n)$ a sequence of positive numbers such that either ...
2
votes
4answers
73 views

Calculate simple integral undefined

I'm not being able to calculate $ \large{\int{\sqrt{\frac{x}{a-x}}} dx} $ , someone could help me? I tryed to use integration by parts, but i achieved $0 = 0$. Thanks in advance.
3
votes
2answers
72 views

Understanding the definition of a Integral

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on ...
1
vote
2answers
25 views

identity on integration's extremes.

We have a continuous functions $f(x)$, variable $x,t\in \mathbb R$ and a real (positive) parameter $r$. Are true the following identity $$\int_0^t (f(r+x)-f(x))\, dx=\int_t^{t+r} f(x)\,dx-\int_0^r ...
2
votes
1answer
45 views

Triple integral and region of integration: $f(x,y,z)=z^2$

Calculate $\iiint_s f(x,y,z) \;dxdydz$ for $f(x,y,z) = z^2$ and $S$ the region bounded by $z=0$, $x^2 + z = 1$ and $y^2 + z=1$ I've already plotted the region but I am really having difficult to find ...