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 ...

learn more… | top users | synonyms (2)

0
votes
0answers
24 views

Examples using LIATE rule for Integration by parts

Popular textbook contain many examples of integrals which can be computed by parts using the LIATE Rule. However there is almost no example of the case LI, that is a logarithmic function times an ...
0
votes
0answers
5 views

Approximation technique of common probability distributions that can be convoluted and integrated fast

I am looking for a approximation technique of functions with two conditions: It is possible to perform a fast approximate convolution with the approximate functions. It is possible to numerically ...
1
vote
2answers
60 views

I Don't Understand This Arc Length Formula

I'm taking the following from my Stewart's Calculus 7E. This is a introductory section of finding arc length. My Problem I follow what they're saying. If we approximate portions of the curve using ...
6
votes
2answers
87 views

What is the function $f(x)=x^x$ called? How do you integrate it?

For real numbers $x > 0$, the function $f(x)=x^x$ seems pretty cool. Is there a name for this function? It's obviously been studied before. It grows faster than exponential functions and ...
0
votes
0answers
7 views

Is it true that x(t)*\delta(t-nT) is nonzero iff n=t/T?

Note: T is just some constant and n is an integer. I'm trying to verify the steps.But I'm unsure that the statement is true since for n=5, the expression would be nonzero at 5T (assuming x(5) is ...
3
votes
0answers
30 views

How to evaluate these indefinite integrals with $\sqrt{1+x^4}$?

These integrals are supposed to have an elementary closed form, but Mathematica only returns something in terms of elliptic integrals. I got them from the book Treatise on Integral Calculus by ...
0
votes
1answer
6 views

Generalized Riemann Integral: Uniform Convergence

Disclaimer This thread is meant to record. See: Answer own Question And it is written as question. Have fun! :) Reference This thread is related to: Generalized Riemann Integral: Nonexample? ...
-1
votes
2answers
40 views

Problem in Spivak Calculus [on hold]

Suppose that $f$ and $g$ are differentiable functions that satisfy $\int_{0}^{f(x)}f(t)g(t)dt=g(f(x))$. Show that $g(0)=0$. Thank you.
1
vote
1answer
38 views

Is This a Valid Way of Finding Apery's Constant?

Is this a valid way to find Apery's constant? Consider the power series generated by $\ln (1-x)$ Let $x= e^{ix}$. Integrate $f(e^{ix})$ two times. Then let $x=\pi$. By doing these manipulations do you ...
1
vote
0answers
13 views

Finding the volume of a cube using spherical coordinates

Calculate the volume of a cube having edge length $a$ by integrating in spherical coordinates. Suppose that the cube have all the edges on the positive semi-axis. Let us divide it by the plane passing ...
0
votes
0answers
31 views

Indefinite Integral of a cube root of a function

What is the integral $$\int\,\sqrt[3]{\vphantom{\arge A}\, x^{2} + 1\,}\,\,{\rm d}x$$ Any hint will suffice .
2
votes
1answer
25 views

Triple Integrals: Conversion

I'm currently in second year calculus and have come across a problem that I'm struggling badly to try and understand. The question is as follows: Sketch the region of integration of the following ...
0
votes
1answer
18 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: ...
1
vote
2answers
39 views

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

How to integrate $\frac{\sqrt{z+1}}{z}$ Anyone could help me? Thanks
0
votes
1answer
52 views

Prove that $F=\int_x^{x^2} \! \frac{\sin t}{t} \, \mathrm{d}t$ is differentiable. [on hold]

Prove that $F=\int_x^{x^2} \! \frac{\sin (t)}{t} \, \mathrm{d}t$ is differentiable and on the interval $(1, \infty)$ and calculate $F'(x)$ for $x\in (1,\infty)$. Thanks in advance!
0
votes
1answer
20 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
20 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
votes
0answers
16 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
28 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
14 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
63 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
49 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
4answers
109 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
33 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
37 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
62 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
45 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
53 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
75 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
87 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
59 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
15 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
46 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
48 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
81 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
30 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
votes
0answers
23 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
70 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
54 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 ...