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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Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
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3answers
22 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
3
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0answers
25 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
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1answer
36 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
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3answers
57 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
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0answers
29 views

Evaluating integral involving product of cosine inverse

I am trying to evaluate the below mentioned integral which involves product of two cosine inverses and two variables $x$ and $y$. I need to evaluate the integral or get an approximate value of this ...
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0answers
11 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
3
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3answers
128 views

Why consider square-integrable functions?

Why are $L^2$ functions important? From reading around I have three hypotheses: they show up in QM (but, why?) they form an inner product space (but, is that a "tight bound" or is the class easily ...
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0answers
45 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
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0answers
47 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
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0answers
45 views

Solution to the Integral

I am trying to solve a pdf which contains the following integral. The integral would involve the inverse of cosine function. Can anybody provide me the method how to solve the below mentioned ...
1
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3answers
45 views

Integration by parts: $\int e^{-\theta}\cos7\theta \;d\theta$

$$\int e^{-\theta}\cos7\theta \;d\theta$$ I started off by using $u=\cos 7\theta$ and$ \;dv=e^{-\theta}d\theta$, however, this just led me in a circle. I am now at: $$u=e^{-\theta},\;dv=\cos 7\theta ...
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2answers
31 views

Calculus 2: Strategy for Integration, Integral of e^(x+e^x)dx

How would you find $\int e^{x+e^x}dx$? I know I need to use $u$-substitution but I tried changing what I use for $u$ but I still could not get the right answer. If someone could push me in the ...
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1answer
40 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
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1answer
33 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
1
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1answer
26 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
0
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1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.
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3answers
234 views

Why we use dummy variables in integral?

I want to know why we use dummy variables in integral? thanks so much.
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2answers
129 views

Why does integration of acceleration data create a slope?

I created a 100hz sine wave in code. When I graph the waveform I get this: When I do an integration on this pure sine wave to get a velocity waveform I get: Is this normal? I do not have a ...
0
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3answers
57 views

Evaluate the integral $\int_0^{1/4}\frac{x-1}{\sqrt{x}-1}\mathrm dx$

so I have this Integral I have to solve without a calculator. $$\int_0^{1/4}\dfrac{x-1}{\sqrt{x}-1}\mathrm dx.$$ How would I go about finding the antiderivative of that fraction?
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1answer
29 views

Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
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0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
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0answers
44 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
2
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1answer
49 views

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$ This looks like an interesting problem with an elegant ...
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1answer
24 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
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0answers
54 views

How can I evaluate this integral?? [duplicate]

integral $\int_{0}^{\infty} \frac{cosx}{x^2+1} dx$? I got the answer is $\frac{\pi}{2e}$ by using Wolfram. But can't do it by myself... need some help
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0answers
28 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
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4answers
65 views

What is the most efficient way to integrate $(x-3)\sqrt{x^2+3x-18}$?

I can do the problem, but it is becoming so big,that I do not feel to do it anymore. Can anyone give the shortest method for this problem? $$\int (x-3)\sqrt{x^2+3x-18}\,dx $$
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1answer
33 views

how to remove modulus signs after integrating

$$ \frac{dy}{dt} + k\frac{t^2 -3t + 2}{t+1}y = 0,\ \ \ \ \ \ \ y(t_0=0)=A>0\\ -\int \frac{k}{y} dy = \int (t-4 + \frac{6}{t+1}) dx $$ After integrating the above how do you express $y$ in terms ...
4
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1answer
49 views

Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies ...
3
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2answers
30 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
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3answers
35 views

integrate $\int e^{-iwt}dt$

I have this integral: $$ \int e^{-iwt}dt$$ I know that $\int e^{kx}=\frac{e^{kx}}{k}$ so therefore the $ \int e^{-iwt}dt$ would be $\frac{e^{-iwt}}{-iw}$ but Wolfram Alpha says that it is $\int ...
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0answers
94 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
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2answers
45 views

Can anybody prove why integral of f*f from 0 to 1 not 0? [on hold]

If I have a function f, which can be all real polynomials, Why integral of f * f on [0,1] is not equal to 0 ? I know intuitively, but I need to see the proof
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2answers
31 views

Integrating this improper integral to test for convergence?

I'm trying to integrate this: $$\int^\infty_0 \frac{8}{\sqrt{e^{x}-x}} \,dx$$ And use the Direct Comparison Test to find out whether it diverges or converges. I looked at a similar problem: and I ...
0
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1answer
72 views

Finding total work by integration

The following tank is completely filled with water. Find the total amount of work done in pumping water out of the outlet. Note that the density of water is 1000 kg/m$^3$ I feel like I am ...
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0answers
8 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
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2answers
60 views

Indefenite Integral requiring substitution

Can someone please help me find a useful substitution for the following integral: $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2}dx$$ I tried letting $ u = \sqrt{x} $ But I couldn't proceed. Please help.
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2answers
63 views

Can I integrate $\frac{x}{1-x}$ by substitution?

I saw a person use substitution like this: $$\int \frac{x}{1-x} dx$$ Let $u= (1-x)$, $x= 1-u, du= -1\cdot dx$ $\Rightarrow$ $-du=dx$ $$\int \frac{1-u}u (-du)$$ Can I use substitution like this? I ...
3
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1answer
49 views

Another parametric integral relating to hyperbolic function

if $0<a\leq1$, then canwe get a closed form of $$I(a)=\int_0^\infty\frac{x}{\tanh x}\frac{1}{\cosh^2(ax)}dx.$$ In fact,if $a=1$,$I(a=1)=\pi^2/8$.
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2answers
50 views

parametric integral relating to hyperbolic function

Suppose that $a$ is real number such that $0<a<1$, how can we calculate $$ I(a)=\int_0^\infty \big(1-\frac{\tanh ax}{\tanh x}\big)dx .$$ As for some speical cases, I can work out $I(1/2)=1$. ...
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2answers
96 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
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3answers
38 views

This is the question about integration. I want to know how to approach this question. [duplicate]

My solution makes same loop, which eventually makes the equation as 0 = 0 form.
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1answer
34 views

Integration by Parts and Convergent/Divergent Series Calculus

We are asked to integrate: $$\int x (lnx) dx$$ Integration by parts gives us: (using L-I-A-T-E) $$u = lnx$$ $$ du = (1/x)dx$$ $$ dv = xdx $$ We find v by integrating dv: $$ v = (1/2)x^2 $$ ...
3
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1answer
40 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
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2answers
45 views

Integration of $1/(x^2+x\sqrt{x})$

The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached. The problem of my solution is if I use partial fraction, loop will be made, and this makes ...
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2answers
44 views

Integration of $(5x^2+2x-5)/(x^3-x)$

The problem is to evaluate $\int \frac{5x^2+2x-5}{x^3-x}\,dx$. This is the solution that I tried: I really have no idea of this problem. After check my solution, if there are any problem that ...
0
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2answers
53 views

This is the question about integration.

My idea is to use substitute integration. Since there is square root of (1-x^2), I made x = cos^2t, and then eliminated square root. I don't know why my answer is wrong. I already conducted ...
4
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0answers
61 views

Closed form for $\int_1^\infty\frac{dx}{\Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{dx}{\Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\Gamma(x)$ or the like) don't ...
0
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6answers
83 views

For polynomials $f,g$, why is $\int_0^\infty \frac{fg}{e^x}\, dx$ absolutely convergent?

Why does the integral $\displaystyle \int_0^\infty \frac{fg}{e^x}\, dx$ have to be convergent for all real polynomials $f$ and $g$? Can anybody give me a proof?