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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38 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
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1answer
46 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
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2answers
80 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 ...
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0answers
33 views

It's question about area of surface of 3 dimensional figure by using integration.

This is my solution. I don't know why the answer is still wrong
4
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0answers
62 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
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1answer
135 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. ...
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1answer
29 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 !
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2answers
101 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
86 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
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2answers
25 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 ...
2
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3answers
81 views

Evaluate the following integral: [closed]

Evaluate: $$\int_{}^{} \frac{\sqrt{a^2-x^2}}{x}\, dx$$ where $a$ is a real parameter and $0<x<a$.
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0answers
13 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 ...
0
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1answer
34 views

Name of special function used used by Wolfram integrator

Integrating $\frac{e^{-r}}{(\sqrt{2t-r})}$ with respect to $r$ between $r=t$ and $r=2t$ on Wolfram (http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605) gives the answer ...
1
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1answer
29 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
45 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 ...
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2answers
22 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 ...
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3answers
44 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 ...
0
votes
0answers
19 views

Differential equation to find account vaule

A retiree deposits S dollars into an account that earns interest at an annually rate r compounded continuously, and annually withdraws W dollars. The account changes at the rate: dV/dt=rV-W Solve this ...
1
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2answers
18 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 ...
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1answer
46 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
28 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,
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2answers
66 views

I would like prove a result in integration

I would like prove this result $$\int_0^1 \frac{\left(\log (1+x)\right)^2}{x}\mathrm dx=\frac{\zeta(3)}{4}$$
11
votes
4answers
146 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 ...
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vote
1answer
38 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 ...
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2answers
84 views

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

$$\int e^{-x^2} dx$$ How do we calculate this integration?
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votes
3answers
297 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 $k \in \mathbb{N}_{>0}$.
0
votes
0answers
31 views

Probability that one uniform distribution is less than another

I am trying to pick the variable r which maximizes an expected return. So I need to calculate the probability that $rn < x$ where n and x are both random variables with uniform distributions ...
1
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1answer
38 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}, ...
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0answers
38 views

How to change specifically the limits of the integral?

Lets say the interval is [a,b]. How then can I switch the interval to [c,d] (without changing the value of the integral)? I can't build the general formula :( please help.
4
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0answers
248 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
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6answers
93 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
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2answers
35 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 ...
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2answers
19 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?) : ...
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0answers
16 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), ...
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0answers
24 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 ...
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1answer
28 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 $[a,b]$.
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1answer
38 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
38 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
20 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
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3answers
287 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 ...
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1answer
39 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 ...
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0answers
26 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
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1answer
18 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
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0answers
38 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
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1answer
46 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) ...
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1answer
32 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
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2answers
95 views

About Integration

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
142 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}
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0answers
37 views

Finding the volume of a solid via triple integrals and spherical coordinates

Let S be the Solid enclosed above by $x^2+y^2+z^2=2$, below by $x^2+y^2=z^2$ and y=0. Use triple integrals and spherical coordinates to evaluate the volume of the solid S.(I just need to know what the ...
2
votes
2answers
36 views

A measure is sigma-finite if, and only if, there exists a integrable function w such that its image is contained in (0,1)

I have to prove the following proposition: Consider a measure space $(S,\Sigma,\mu)$. Prove that $\mu$ is $\sigma$-finite if, and only if, there exists $w\in\mathcal{L}^1(S,\Sigma,\mu)$ such that ...