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 ...

learn more… | top users | synonyms (3)

0
votes
1answer
50 views

Integral of exponent $\iint\limits_0^\infty e^{(t_1 x+t_2 y -y) }\ dy\ dx$

Please help me to solve this equation. I have attempted to answer this one however I always arrived at the wrong answer. Instead of having the positive sign, I always ended with its negative answer. ...
-1
votes
1answer
56 views

Double Integral to Polar Coordinates

Evaluate $$\int_{0}^{2}\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2}dydx$$ by converting to polar coordinates. I sketch the region which is a half circle from $0$ to $2$ on the $x$-axis and $0$ to $1$ ...
1
vote
2answers
61 views

find $\displaystyle \int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$

find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$ If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
3
votes
2answers
86 views

When $\lim_{n\rightarrow\infty}\int_0^1f(x)\sin(nx)=0$

What is a good sufficient condition for $f:[0,1]\rightarrow\mathbb{R}$ such that: $$\lim_{n\rightarrow\infty}\int_0^1f(x)\sin(nx)dx=0$$ If $f$ is differentiable then by integral by part we can ...
0
votes
0answers
35 views

n integrals in summation

I've seen it might be possible to write a summation that looks like $$ \sum\limits_{i=1}^{\infty}\left\{\frac{\partial}{\partial x_i}\left(\frac{xy}{\sqrt{x^2+y^2}}\right)\right\} $$ But what about ...
2
votes
2answers
125 views

Easier method to evaluate integral

This question is worth two marks and IMHO (by the method I use) requires too much work to be worth only two marks. So I am wondering if there is a faster method to evaluate: ...
0
votes
1answer
40 views

Solving an Integral - $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt $

How do we solve $ \int t^2\frac{\left(2t\sqrt{at^2+bt+c} \right )^{2k}}{(at^2+bt+c)} \ dt \tag 1 $ to a finite form? $k,a,b,c$ are constants $at^2+bt+c$ does not guarantee equal roots always
1
vote
1answer
58 views

How to integrate Gravitational force?

The gravitational force is given by $$ F = \dfrac{-Gm_1m_2}{r^2} $$ But, since F = ma, then for an object placed at r distance away from the centre of the earth it would experience $$ a = ...
4
votes
2answers
84 views

How to calculate the value of the special integral

I get $${\left. \frac{\partial ^2}{\partial n^2} \left( \frac{\partial ^2}{\partial m^2} B(m,n) \right) \right|_{m = \frac{1}{2},n = 0}} = \int_0^1 \frac{\ln^2 x \ln^2 (1 - x)}{\sqrt x (1 - x)} \, dx ...
4
votes
0answers
96 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
2
votes
1answer
146 views

Closed-form of $\int_{a}^{b}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx$ for some $a<b$

In this question I asked to prove that $$\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}.$$ If we take a look at the plot of the integrand, then we could see some symmetry-property. ...
-1
votes
1answer
34 views

Help me to solve this question! It's about figuring out the area by using integration

My book said "Use integral 2 pi x ds but I have no idea about ds. Help me to solve this !
6
votes
2answers
133 views

Log integrals I

In this example the value of the integral \begin{align} I_{3} = \int_{0}^{1} \frac{\ln^{3}(1+x)}{x} \, dx \end{align} was derived. The purpose of this question is to determine the value of the more ...
6
votes
3answers
95 views

Integration of $\int (\frac{1-x}{1+x})^{\frac{1}{3}}$

I've been trying to integrate this for a long time but can't. $$\int \left(\frac{1-x}{1+x}\right)^{\frac{1}{3}}$$ I tried assuming $\frac{1-x}{1+x}=t^3$ , also tried integration by parts but it ...
0
votes
2answers
79 views

Negative area below x-axis and above $f(x)$

I was finding the area below the x-axis and above $y = x^2 - 4x$. My outcome was a negative number (fair enough, it's under the x-axis), but wolframalpha for instance gives me a positive number (I get ...
1
vote
3answers
101 views

Evaluate the following integral: [closed]

Evaluate: $$\int_{}^{} \frac{\sqrt{a^2-x^2}}{x}\, dx$$ where $a$ is a real parameter and $0<x<a$.
0
votes
0answers
23 views

Integration : Green's Function in estimating displacement of non-prismatic beams

I'm working on a non-prismatic structure similar to that in Figure 3 of Page 10 (345) from an article entitled: "Green’s function for the deflection of non-prismatic simply supported beams by an ...
1
vote
2answers
89 views

Name of special function used by Wolfram integrator

Integrating $e^{-r}/\sqrt{2t-r}$ with respect to $r$ between $r=t$ and $r=2t$ using this widget gives the answer $2e^{-t}F(\sqrt{t})$. However the widget doesn't say what $F$ is. I have looked on ...
1
vote
1answer
31 views

Integrating the Schechter function…

I'm trying to integrate this equation over all L. I really have no idea where to start for some reason :S $$\phi(L)dL=\phi_0\left(\dfrac L{L\star}\right)^\alpha\exp\left(-\dfrac ...
1
vote
1answer
52 views

Evaluate $\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $

I have been trying to evaluate the following integral $$\int_0^\infty \frac{1}{x^{n+2}} c^{1/x} \exp\{-\lambda/x\} \mathrm{dx} $$ What I am getting is $$\frac{1}{\left(\lambda-logc ...
1
vote
2answers
33 views

Riemann sum for unbounded functions

Suppose that $f$ is a Lebesgue integrable function on $[0,1]$ whose set of discontinuities is of Lebesgue measure zero. Is it true that the Riemann sum $\frac1n \sum_{k=1}^n f(k/n)$ converges to ...
0
votes
3answers
49 views

Integral $\int_t^T\frac{1}{\phi-\psi e^{-\gamma(T-s)}} \operatorname d \!s$

I am having trouble solving the following integral: $$ \int_t^T\frac{1}{\phi-\psi e^{-\gamma(T-s)}}ds $$ Maybe there's something obvious I am missing, my approach so far has been to use a ...
1
vote
2answers
22 views

Confused About Trigonometric Substitution

I'm learning Trigonometric Substitutions, they gave us the following example in the book: I'm confused about how exactly we make the substitution $x= a\sin(\theta)$ In regular substitution we have ...
-1
votes
1answer
61 views

Integral involving the error function

Is there a closed form solution to the integrals \begin{align} I_{c} &= \int_{0}^{\infty} \cos(a x) \, \operatorname{erf}(b x) \, dx \\ I_{s} &= \int_{0}^{\infty} \sin(a x) \, ...
0
votes
1answer
104 views

Calculus 7th Ed (Stewart) - Chapter 4 solution 2 page 332

This can be really ridiculous for you but I can't understand why dx is up on the root in solution 2 Shouldn't be "du = root(2x+1)*dx" instead of what is show below? Best Regards,
2
votes
2answers
70 views

I would like prove a result in integration

I would like prove this result $$\int_0^1 \frac{\left(\log (1+x)\right)^2}{x}\mathrm dx=\frac{\zeta(3)}{4}$$
10
votes
4answers
179 views

A closed form for $\int_0^1 \frac{\left(\log (1+x)\right)^3}{x}dx$?

I would like some help to find a closed form for the following integral:$$\int_0^1 \frac{\left(\log (1+x)\right)^3}{x}dx $$ I was told it could be calculated in a closed form. I've already proved that ...
1
vote
1answer
68 views

Evaluating an indefinite integral with an inverse trigonometric function

I'm really stumped on a homework problem asking me to evaluate $\int \frac{ln\ 6x\ sin^{-1}(ln6x)}{x}dx$, and after a few hours of trying different approaches I'd definitely be appreciative for a bump ...
1
vote
2answers
93 views

How to $\int e^{-x^2} dx$

$$\int e^{-x^2} dx$$ How do we calculate this integration?
9
votes
3answers
354 views

Evaluate $ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos (2x)}\right) \mathrm{d}x $

Evaluate the following definite integral: $$ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos(2x)}\right) \mathrm{d}x, $$ where $k \in \mathbb{N}_{>0}$.
1
vote
1answer
40 views

$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Prove that there exists $C > 0$ such that the following implication holds: If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$, then there exists $ \{z_{j_1}, ...
5
votes
0answers
262 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
0
votes
6answers
97 views

Integral $\int_2^4 \sqrt{16-x^2} \operatorname d\!x $

We want the following integral: $$\int_2^4 \sqrt{16-x^2} \operatorname d\!x $$ This is of course part of a circle of radius 4. I was wondering how you can find the area of that part of the circle, ...
1
vote
2answers
37 views

The integral of $x^3/(x^2+4x+3)$

I'm stumped in solving this problem. Every time I integrate by first dividing the $x^3$ by $x^2+4x+3$ and then integrating $x- \frac{4x^2-3}{x+3)(x+1)}$ using partial fractions, I keep getting the ...
0
votes
2answers
25 views

Integral $\int_{-\infty}^0 e^{(-3i+\omega)t} $

Let's say I am integrating this function: $e^{(-3i+\omega)t}$ from $t=-\infty $ to $t=0$ [Note: $\omega$ is just a constant] The same function could be rewritten in this form(i believe?) : ...
0
votes
0answers
32 views

How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), ...
0
votes
0answers
31 views

Is there a finite set comprising the solutions to indefinite integrals of common functions?

There are some integrals that are impossible to express in terms of elementary function, for example, $ \int \frac{e^x}{x} dx $ is only expressible as a "special" function $Ei(x)$, the exponential ...
0
votes
1answer
33 views

On Rieman integral

Suppose $0\leq f$ on $[a,b]$. Could we deduce from $f^2(x)\in R[a,b]$ that $f(x)\in R[a,b]$, where $R[a,b]$ is the set of all functions that are Riemann integrable on $[a,b]$.
1
vote
1answer
47 views

Calculating improper integral

Does anyone know how to solve the following integral: $$I =\int_{0}^\infty \cos(t \mathrm{log}( x))\,\mathrm{e}^{-ax}\, \mathrm{d}x,$$ where $t$ and $a$ are real. Please show some intermediate ...
2
votes
1answer
51 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
3
votes
1answer
29 views

A pair of integrals of rational powers of sines

I'm currently teaching an introductory calculus course which goes through various "techniques of integration." On the way to showing that we can integrate $$ \int R(x, \sqrt{ax^2 + bx + c})dx $$ for a ...
12
votes
3answers
420 views

Integral $\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}}\operatorname d \!x$

Could you please help me with this integral? $$\int_0^1\frac{x^{42}}{\sqrt{x^4-x^2+1}} \operatorname d \!x$$ Update: user153012 posted a result given by a computer that contains scary Appel ...
0
votes
1answer
45 views

Solving $\int \frac{1}{x-1}dx$ in two ways.

I have some confusion with this integral $$\int \frac{1}{x-1}dx$$ I can see the solution is $ln(x-1)$ However if I multiply the top and bottom by $-1$ I get $$\int \frac{-1}{1-x}dx$$ And then ...
-1
votes
0answers
39 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
0
votes
1answer
43 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
2
votes
0answers
39 views

Is it possible to abstract a Riemann integral into a “higher” integral with measure?

I'm not very comfortable with more generalised integrals such as the Lebesgue integral yet, but I'm working through some material to achieve that goal. I have a question which stems simply from ...
2
votes
1answer
50 views

Proving the equality of a sum and integral.

Taken from Rudin's Real and Complex Analysis text: Suppose $f$ is a continuous function on $\mathbb{R}^1$ with period $1$. Prove that $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n\alpha) ...
0
votes
1answer
40 views

Continuity and norm of a functional

Let $E = \mathbb{R} [X]$ equipped with the norm $||p|| = \int_0^1 (|p(t)| + |p'(t)|) \ d t $. Check if the functional $\psi : E \ni p \rightarrow p(0) \in \mathbb{R}$ is continuous, and if it is, ...
5
votes
2answers
114 views

About Integration $ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $

How to calculate the following integral $$ \int \frac{\tanh(\sqrt{1+z^2})}{\sqrt{1+z^2}}dz $$ Is there any ways to calculate those integral in analytic? (Is $[0,\infty]$, case the integral is ...
7
votes
2answers
163 views

A Sine integral: problem I

Is it possible to demonstrate a solution for the integral \begin{align} \int_{0}^{\infty} x^{n} \, \sin\left( a x^{2} + \frac{b}{x^{2}} \right) \, dx \end{align}