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 ...

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16 views

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ?

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ? ( where $R[a,b]$ is the set of all riemann integrable functions on $[a,b]$ )
3
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4answers
113 views

How to integrate $\int \frac{4}{x\sqrt{x^2-1}}dx$

In order to solve the following integral: $$\int \frac{4}{x\sqrt{x^2-1}}dx$$ I tried different things such as getting $u = x^2 + 1$, $u=x^2$ but it seems that it does not work. I also tried moving ...
-1
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1answer
39 views

How to find the antiderivative of f(x).

While studying, I learned that the antiderivative of $1/f(x)$ is simply ln$\lvert f(x)\rvert$. Why is this so?
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1answer
34 views

Help on finding the closed form of the integral

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$ where ...
1
vote
1answer
36 views

Integrals, intermediate value theorem question

f∈c[a,b] (f is continuous in [a,b]), prove: We tried to use the integral intermediate value theorem to try to prove it but we don't understand why the limit has to be the max and not any other value ...
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2answers
49 views

Integral for cos

$$\int_{-a}^{+a}A^2\cos^2\left(\frac{n\pi a}{2}x\right)dx$$ where $A, n, \pi,a$ are constants. I was thinking about trigonometric formula for $\cos^2x=\frac{1+\cos2x}{2}$
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1answer
26 views

Area surrounded by a curve

I would need help to calculate the area surrounded by a curve. The curve is given with the following polar coordinates: I know we need need to integrate with respect to r and theta but am stuck ...
2
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0answers
21 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
2
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4answers
52 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{3}{8}{(\dfrac{\pi}{\alpha^5})}^\dfrac{1}{2}$

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
0
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1answer
22 views

Why is $F'(x) = 2x·\tan(x^2)-\tan x$ if $F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$?

Evaluate $F'(x)$ if $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$$ I tried to do this by the change of variables formula and hence, $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du=\int_{\sqrt x}^{x}\tan ...
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vote
1answer
60 views

Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

Can anyone help me find the following indefinite integral: $$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$ I cannot think of any transformation...
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1answer
46 views

Definite integral: $\int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx$

The goal is to solve this: $$ \int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx $$ with $a>0$. Really not sure how to attack this one. The integrand seems to be capable of ...
2
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5answers
51 views

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
3
votes
1answer
23 views

How to prove define integrate from f(sin x)

i need help for prove this problem , i dont have idea for this prove, i very appreciate your sugerences. $$ \int ^{\pi }_{0}xf(\sin x)\,dx = \int ^{\pi }_{0}\frac{\pi }{2} f(\sin x)\,dx $$
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0answers
18 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
5
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1answer
52 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
0
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0answers
35 views

why do we need integration in the probability or measure theory? [on hold]

As the title is, why do we need integration in the probability or measure theory? because we do not learn how to calculate an area or volume under some function in the field. Some practical examples ...
4
votes
4answers
114 views

Find $\int_0^1(\ln x)^n\hspace{1mm}dx$

I am not a big fan of induction, it's just a personal preference. Is there a method other than induction. Answer is $n!$ by the way
3
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0answers
43 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
6
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0answers
56 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
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1answer
42 views

Easy method to check integrability as elementary functions

What could be an easy method (Calc 1) to check if a given integral is not integrably in terms of elementary functions? Take for example: $$ \int e^{-t^{2}}dt$$
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0answers
84 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
-4
votes
0answers
49 views

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$? [on hold]

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$?
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2answers
55 views

Volume of Solid Enclosed by an Equation

I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple ...
10
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1answer
495 views

The word “integral” in calculus unrelated to “integral” / “integer” in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
1
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2answers
45 views

How to evaluate the line integral $\int_C (y-z)\,dx+(z-x)\,dy+(x-y)\,dz$

How to evaluate the line integral $\int_C (y-z)\,dx(z-x)\,dy(x-y)\,dz$. The curve $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x-z=1$. I am really stuck on how to to do this ...
0
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1answer
34 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
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vote
4answers
36 views

Find equation of curve

${dy \over dx}= (3x^2-a)^2$, where $a$ is a constant. Given that the curve has a stationary point at $(3,2)$, find the equation of the curve. I managed to get the equation $y=3x^3+3ax^2+xa^2$+c. I'm ...
0
votes
2answers
74 views

Proving that a function is Riemann Integrable

The usual definition to the Riemann integral is: for every $ε>0$, there exists $\delta$ such that if $P$ is a partition of $[a,b]$, and $\|P\|<\delta$, then $|S(f;P)-s|<\epsilon$. Then $f$ is ...
0
votes
1answer
45 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
2
votes
3answers
43 views

Find expression in terms of x

Knowing that $$\frac{dy}{dx}= k\cdot x^{\frac{1}{3}}$$ and given that it passes through points $(1,4)$ and $(8,16)$, find an expression for the path in terms of $x$. I found out that $$y= \frac34 k ...
0
votes
1answer
28 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...
1
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1answer
44 views

Reasons for different answers when finding area using Simpsons rule and numerical integration?

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
2
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0answers
31 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
0
votes
2answers
83 views

Evaluating $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates

I'm having issues solving $\iiint_v(3x^2+3y^2+3z^2) \, dv$ using Spherical Coordinates I made the ffg substitutions: $x=r\sin\theta\sin\phi, y=r\sin\theta \cos\phi, z=r\cos\theta$ Thus ...
2
votes
4answers
75 views

Solve $\int\frac{8x+9}{(2x+1)^3}\,dx$.

Do I split $\displaystyle\int\frac{8x+9}{(2x+1)^3}\,dx$ into partial fractions? Or do I use $(2x+1)^{-3}$ by itself? Not sure what to do. Please advice. The answer given is ...
0
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2answers
45 views

How is $ \frac{\sqrt{a}}{a+1} (0^{a+1}+1^{a+1}) $ equal to $ \frac{\sqrt{a}}{a+1} (-1)^a $

I am trying to integrate this equation $$ y = \int_{-1}^0 \sqrt{a} x^{a} $$ $$ y = \sqrt{a} \int_{-1}^0 x^{a} $$ $$ y = \frac{\sqrt{a}}{a+1} \int_{-1}^0 x^{a+1} $$ $$ y = \frac{\sqrt{a}}{a+1} ...
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1answer
26 views

Find value of define integral with [on hold]

Hi i need help for this problem, i very appreciate your sugerences. $$F(x)\text{=}\int ^{g(x)}_{0}\frac{dt}{\sqrt{1+t^{2}} } $$ And $$g(x)\text{=}\int ^{\cos x}_{0}[1+\sin t^{2}]dt$$ For $F'(π/2)$.
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2answers
356 views

demostration of interger part integration.

I need help for solving this demostration, I appreciate your suggestions very much. $$\begin{array}{rclr} \int ^{n}_{0}[x] dx= \frac{n(n-1)}{2} \end{array}$$ Pd. If you have any suggestion of a ...
0
votes
1answer
52 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
2
votes
2answers
28 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
8
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1answer
87 views

Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

I'm just wondering if the following integration is valid. \begin{array}{l} \int {\frac{1}{{\sqrt {{a^2} + {x^2}} }}} dx\\ {\rm{Let }}{u^2} = {a^2} + {x^2}\\ 2udu = 2xdx\\ \frac{{du}}{x} = ...
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0answers
20 views

How do you find the volume of a function rotated about the x axis along it's derivative? [on hold]

So when you rotate a function, it is usually vertically. How do you rotate it around it's derivative, assuming that volume/area can overlap?
4
votes
4answers
91 views

Am I getting the right answer for the integral $I_n= \int_0^1 \frac{x^n}{\sqrt {x^3+1}}\, dx$?

Let $I_n= \int_0^1 \dfrac{x^n}{\sqrt {x^3+1}}\, dx$. Show that $(2n-1)I_n+2(n-2)I_{n-3}=2 \sqrt 2$ for all $n \ge 3$. Then compute $I_8$. I get an answer for $I_8={{2 \sqrt 2} \over 135}(25-16 ...
-4
votes
2answers
28 views

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $H(x) := \int_x^b f \ \ ,x\in I.$ [on hold]

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $$H(x) := \int_x^b f, \ \ x\in I.$$ To find $H'(x)$ for $x \in I.$ I am stuck with the problem please help.
1
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0answers
33 views

Integral of a function which is everywhere discontinuous?

Yesterday, I tried to carry out a little thought experiment when it came to taking limits and have found that it has pushed my understanding of them to the breaking point. I tried considering the ...
3
votes
2answers
72 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
-2
votes
0answers
21 views

Integral Proof: Integral between h and 0, (h - z)(z - l)dz [on hold]

How does the following give the result of l = h/3: Integral between h and 0, (h - z)(z - l)dz
1
vote
1answer
40 views

Double Integration word problem

In a certain metropolitan area, the population is approximated by the function: $$P(x,t)=\frac{\ 7274e^{0.5t}}{1+x}$$ Where $x$ is the number of miles from the center of the city, and $t$ is the ...
1
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0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...