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
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1answer
11 views

Integral of absolute value

I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: ...
0
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2answers
22 views

How to integrate $\frac{\sqrt{z+1}}{z}$

How to integrate $\frac{\sqrt{z+1}}{z}$ Anyone could help me? Thanks
-1
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1answer
33 views

Prove that $F=\int_{x^2}^x \! \frac{sin t}{t} \, \mathrm{d}t$ is differentiable.

Prove that $F=\int_{x^2}^x \! \frac{sin (t)}{t} \, \mathrm{d}t$ is differentiable and on the interval $(1, \infty)$ and calculate $F'(x)$ for $x\in (1,\infty).$ Thanks in advance!
0
votes
1answer
19 views

The best constant in an integral inequality

I find a interesting inequality. Suppose that $y=y(x)$ is a differentiable function in $(0,L)$ and $y(0)=y(a)=0$. Consider the fraction $$ F[y]=\frac{\int_0^{L}\vert y'\vert^2dx}{\int_0^L\vert ...
0
votes
0answers
17 views

How can i solve this integral which involves complex number?

Is there anyone able to solve the following integral? $\int_{z=0}^\infty(c-iz)^{-s-1}*e^{-z^{\alpha}*e^{(i\beta\alpha \frac{\pi}{2})}}dz$ Thanks
0
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0answers
15 views

Rotation of the integration contour through an angle

$\int_{i=0}^\infty ({ue^\frac{ir\pi}{2\alpha})}^{-s}*e^{({{-u^\alpha}e^\frac{-ir\pi}{2})}}\frac{du}{u} $ From this integral, i have to rotate the integration contour through $\frac{-r\pi}{2\alpha}$ ...
0
votes
1answer
25 views

Explaining the signs of given solution using fundamental theorem of calculus

Assume that $y=f_1(t)$ and $y=f_2(t)$ are two solutions of the following function: $$\frac{\mathrm d y}{\mathrm d t}=\mathrm e^{t^3}- \mathrm e^{t^4}$$ and $f_1(0)>f_2(0)$. How can I describe the ...
0
votes
0answers
13 views

how to graph on laptop? draw any shapes and draw an imaginary line or curve? [on hold]

help me guys i'm writing my seminar paper and I'm lack on computer technology. Thanks
4
votes
0answers
56 views

How can we evaluate this tough integral?

$$ \int \frac{\sqrt{\sin\sqrt x}\cos \sqrt x}{1+x^2} dx $$ I have tried combinations of $x=t^2$, integration by parts, $\tan\left(\dfrac u2\right)$ substitutions it got even more complicated. Is ...
0
votes
0answers
15 views

Split this integral

I need to split this integral if possible: \begin{equation} \int_{\mathbb{R}^d} e^{\sum_{i=1}^dx_iz_i}cos(\sum_{i=1}^dy_iz_i)d\mathbf{z} \end{equation} I wanted split into two part : one with $x_i$ ...
-4
votes
1answer
47 views

Integration problem [on hold]

Integrate the following using basic rules of integration in physics: $$\int_0^{\pi/2} \sin t \cos t ~\textrm{dt}$$
4
votes
3answers
91 views

How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Show that this following integral is divergent (or diverges, if you prefer) $$\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$$ I know when $x=0,1$ are singularities of the function and I want use this ...
1
vote
1answer
29 views

Question about substitution method in integration

It is common that we replace $\int u(x)v′(x)\mathrm{d}x$ by $\int u \mathrm{d} v$ where both $u$ and $v$ are continuous functions of $x$. My question is, must we ensure that $u$ can be written as a ...
-2
votes
0answers
27 views

Solving an Integral by Summation [on hold]

My final answer for this question was 90 but I'm not quite sure if I'm even doing it right... I was wondering if anyone could help me solve this for me to check my work against. ...
0
votes
1answer
34 views

Integrand for a set of points

I need help finding what I should be integrating when the question asks to find the double integral to find the volume of the tetrahedron given the points $(0,0,0),(3,0,0),(2,1,0),(3,0,4)$. Would the ...
3
votes
2answers
57 views

Using an Integral to Solve for a Variable a

I am struggling to use the following equation: $$ \int_0^a \sqrt{a^2-x^2}\,\,\text{sgn}(|x|-1)\, dx = 0 $$ where $a > 1$, to deduce that $a = \text{cosec}(\frac{\pi}{4} - \frac{\alpha}{2})$, ...
0
votes
1answer
26 views

Function of a surface area?

$$\iint\limits_s {y \cdot dS}$$ $$z=x+y^2$$ $0 \le x \le 1$ and $0 \le y \le 2$ if you graph $z$, then you get a surface in a 3d scalar field from the ranges above. How does just $y$ in the ...
3
votes
0answers
40 views

Fubini's theorem application proof check

I have proven a problem but I am unsure whether it is correct because the proof seems so simple that I think I might be mistaken. Please be kind to comment on my proof and tell me whats wrong with it. ...
0
votes
2answers
52 views

How do you apply $u$-substitution to the integral $\int v\sqrt{2v^2+1}\,dv$?

When doing $u$-substitution of the following integral problem, does the $v$ disappear because it has a value of $1$? $$\int v\sqrt{2v^2+1}\,dv$$
3
votes
1answer
72 views

Taylor series of a definite integral

Consider the function of a parameter $\alpha > 0$, given by $$f(\alpha) = \frac{2}{\sqrt 2\pi} \int_0^\infty e^{\dfrac{-x^2}{2\alpha^2}}\cosh(x)\log\cosh(x) dx.$$ From Wolfram-alpha, it seems that ...
0
votes
0answers
14 views

Show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a potential function

Let $F=(f_1,f_2)$ be conservative over the open rectangle: $$R=\{(x,y):|x-a|<r,|y-b|<r\} $$ I need to show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a ...
2
votes
4answers
85 views

How to integrate $1/(u^2 + u^4)$ du?

I did a trig substitution with $x = \tan \theta$ followed by a regular $u$ substitution and I got the integral down to $$\int \frac1{u^2 + u^4}\mathrm du$$I just need a reminder of what this would be ...
2
votes
5answers
58 views

How to integrate $\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$?

How to integrate $$\int_1^\infty \frac{dx}{x^2\sqrt{x^2-1}}$$ I tried both $t=\sqrt{x^2-1}$ and $t=\sin x$ but didn't reach the right result.
0
votes
0answers
14 views

Minimization of the integral with respect to a parameter

Intro Let $f$ be a a real-valued function parametrized by a parameter $\alpha \in \mathbb{R}$ and let $J\colon \mathbb{R} \to \mathbb{R}$ be a functional defined as follows: $$J(\alpha) = ...
1
vote
2answers
43 views

Solve this indefinite integral, based on a volume problem

This is making me extremelly pissed off, because I saw a similliar integral that was apparently unsolvable, and now dear prof send this in the list without any resolution or help. The whole question ...
2
votes
1answer
47 views

Prove or disprove following integral.

Assume $L$ a constant, and assume $x$ real. Is the following equation true? $$ \int_{-\infty}^\infty\frac{1}{k^2}\exp(-ikx)dk = \frac{L}{|x|} $$ If it is true, find the value of $L$. If it is not, ...
3
votes
3answers
78 views

Find $\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$

Find $$\int_0^2 \arctan(\pi x)-\arctan(x)\, \mathrm dx$$ The hint is also given : Re-write the Integrand as an Integral I think we have to Re-write this single integral as a double integral and ...
2
votes
1answer
28 views

Complex integral and Laurent series

Could you help with solving this complex integral: $$\int_C z^3\exp{\left(\dfrac{-1}{z^2}\right)} dz$$ where $C$ is $|z|=5$. I am expecting that the Residue Theorem will be needed. The answer should ...
1
vote
1answer
44 views

Integration (Cosine Function)

Ive been doing some integration study and ive been caught by this question. Anyone have any ideas? Thanks. Apologies on how the question is presented, im no quite sure how to do it properly yet. - ...
0
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0answers
22 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
0
votes
3answers
69 views

Example of Riemann integrable $f: [0,1] \to \mathbb R $ whose set of discontinuity points is an uncountable and dense set in $[0,1]$ [on hold]

Give example of a function $f: [0,1] \to \mathbb R $ which is integrable ( Lebesgue or Riemann , if possible , both) but whose set of discontinuity points is an uncountable set and dense in $[0,1]$ ...
-3
votes
0answers
24 views

Solving second order ordinary differential equations [on hold]

I would like you to ask you to solve the following two ordinary differential equations. It is highly appreciated your kind consideration in advance. 1.Solve y"(x)=f(x) 0<=x<=1, where f is ...
0
votes
1answer
20 views

prove integration formula relating to derivatives

Could any one help me solve this problem ? it is from Apostol's calculus volume 1
7
votes
0answers
49 views

Hard sum with harmonics numbers

Prove or disprove that $S=\displaystyle\sum_{n=1}^{\infty}\frac{{H_n^{2}}~{H_n^{(2)}}+3{H_n^{(4)}}}{n~2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$.
1
vote
1answer
59 views

$ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $?

I have the following implication, and I need to determine whether it's true: $ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $ I tried solving ...
1
vote
0answers
39 views

$F(x) = \int_0^x f(t)~dt \implies F(1)=f(0)+\int_0^1(1-t)f'(t)~dt$?

f is differentiable and has a continuous derviative, and $F(x) = \int_0^x f(t)~dt$. Based on this assumption, I have the following statement which I need to determine whether it's true or false: ...
1
vote
2answers
53 views

f is even or odd, prove that f^2 is even

I need to verify whether a statement is correct or false. The statement is as following: If the function f is either odd or even, then the function f^2 is even. To my understanding, the statement is ...
0
votes
0answers
20 views

big $\mathcal O$ for number of prime in an interval?

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+\mathcal O(\sqrt x \log x)$$ I am trying to understand how to deal with the ...
0
votes
0answers
18 views

Integrals: Average(f)*Average(g)=Average(f*g) [on hold]

So I've got everything but question #3 here. I understand that it isn't simply (1/4)(1/4)=16. And also not (1/4)(1/4)(1/4)=1/64. But I can't think of what else it might be. It isn't discussed in the ...
0
votes
2answers
56 views

Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
-1
votes
1answer
49 views

Integration by Substitution, can't solve (Working Added )

My Working: $$\displaystyle dx = du/2x$$ sub dx and U into equation $$ x^2 \int x(U)^{3/2} du/2x$$ Eliminate x $$ x^2/2 \int (U)^{3/2} du$$ $$ x^2/2. [2(U)^{5/2}/5]$$ then $$ ...
0
votes
4answers
69 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
0
votes
0answers
9 views

Recursive formula for Laguerre guassian integral?

The integral of interest is: $ I_{l, m} = \int_{u0}^{u1} u^{(l+1)/2} e^{-u/2} L_m^l(u) du $ where $L_m^l$ is the laguerre polynomial. What I'm interested in is getting some relation to lower order ...
3
votes
3answers
231 views

Indefinite integral of a simple function

$\int 2(1 + \tan^2 x)$ My work : $2(1 + \tan^2(x) = 2 + 2\tan^2x$ $2x + \frac{2}{3}$ $\tan^3(x) \cdot \ln|sec(x)| + C$ The answer says no, after multiple tries :(
0
votes
1answer
34 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
3
votes
4answers
77 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
2
votes
1answer
13 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
3
votes
1answer
72 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
3
votes
2answers
81 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
2
votes
4answers
52 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?