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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1answer
43 views

Most efficient way to integrate $\int_0^\pi \sqrt{4\sin^2 x - 4\sin x + 1}\,dx$?

$$\int_0^\pi \sqrt{4\sin^2 x - 4\sin x + 1}\,dx$$ Please help with this. I cannot do this problem in a definite way.
0
votes
1answer
16 views

Estimating area under curve with 6 and then 12 rectangles [on hold]

Estimate the area under the graph of $$f(x) = 3 x^3 + 4$$ from $x = -1$ to $x = 5$, first using $6$ approximating rectangles and right endpoints, and then improving your estimate using $12$ ...
-1
votes
1answer
12 views

Triple integration in cylindrical coordinates (obtaining the formula to integrate)

Can someone help me out with this? For some odd reason, I am able to derive the boundaries but cannot figure out the formula to use. I thought it was r^2 but apparently it's wrong.
1
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0answers
6 views

Dyson-expansion like multidimensional integral

Let $n \ge 1$ be an integer. Now let $0 \le t_0 \le t$ and $\beta \neq 1$ be real numbers. Now, let $\vec{p} := (p_0,p_1,\cdots,p_n)$ be strictly positive integers. Also let $(x)_{(n)} := x(x-1)\cdot ...
0
votes
2answers
29 views

Is this a quadratic factor or a repeated linear factor?

If I need to integrate something like $\frac{1}{y^2(1-y)}$, and I use the method of partial fractions, how do I know whether the $y^2$ in the denominator is a quadratic factor or just a repeated ...
2
votes
1answer
34 views

indefinite integrals equal imply integrands equal?

if there is an indefinite integral equality does it mean that there is an integrand equality? $$\int f(x)\,dx = \int h(x)\,dx \quad \overset{?}{\Longrightarrow} \quad f(x) = h(x)$$ I know that $$\int ...
-1
votes
1answer
54 views

Compute$\int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$. [on hold]

Compute: $\displaystyle \int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$.
1
vote
3answers
71 views

Prove $f(x)\equiv C$

$f(x)\in C[a,b]$.For any $g(x) \in C[a,b]$ ,which has the property that $\int_a^b g(x) dx=0$,$\int_a^b f(x)g(x) dx=0$. Prove:$f(x)\equiv C$,$C$ is a constant. I haven't any ideas yet. I'm thinking ...
0
votes
0answers
13 views

Calculate the volume of a cube in spherical coordinates

I need to calculate the volume of a cube having edge length $a$ by integrating in spherical coordinates. Any help?
0
votes
1answer
33 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...
1
vote
2answers
36 views

Uniqueness Proof for solution to $\nabla^2 G(\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| \rightarrow \infty$

I'm having difficulty understanding the derivation of solution to this equation: $\nabla^2 G (\textbf{r}) = \delta(\textbf{r})$ with $G \rightarrow 0$ when $|\textbf{r}| -> \infty$ in $R^n$ where ...
1
vote
2answers
54 views

Computing the limit of a summation of sequence

How to compute the limit $$\lim _{n\rightarrow \infty }\left( \dfrac {1}{\sqrt {n^2+1^2}}+\dfrac {1}{\sqrt {n^2+2^2}}+\cdots+\dfrac {1}{\sqrt{n^2+n^2}}\right)$$ The answer is $$\ln ( \sqrt{2}+1)$$ ...
12
votes
4answers
597 views

Can all real polynomials be factored into quadratic and linear factors?

So I understand how to do integration on rational functions with a linear and a quadratic denominator, and I understand how to do a partial fraction decomposition, but I was wondering what happens if ...
4
votes
5answers
106 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
4
votes
2answers
84 views

How to do integration of this?

$$\int_0^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx\quad\quad a > b > 0$$ I have no idea how to compute this. Any help would be greatly appreciated.
1
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0answers
33 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
7 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
63 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
100 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
10 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 ...
5
votes
0answers
41 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
11 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? ...
0
votes
2answers
41 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
44 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 ...
2
votes
0answers
14 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
36 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
28 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
20 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
43 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
54 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
21 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
22 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
18 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
33 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
71 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
16 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
50 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
120 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
35 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 ...
0
votes
1answer
38 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
68 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
49 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
79 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) = ...