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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19 views

Example of a Riemann integrable sequence of functions such that the the sequence of Riemann integrals diverges but… (see below)?

Is there a sequence $(f_n)$ of Riemann integrable functions such that $\lim f_n(x) = f(x)$ almost everywhere on $[a,b]$ and $\lim\int_a^bf_n$ does not exists in Riemann sense, but it does in Lebesgue ...
2
votes
2answers
52 views

Recursive integration

The integral I have is $$I_{n} = \int^{\pi/2}_{0} \cos^{2n+1}y \ \mathrm{d}y$$ And I have found $I_{n} = \frac{2n}{1+2n}I_{n-1}$ but I want to express $I_{n}$ in a form without $I_{n-1}$ how do I do ...
4
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2answers
67 views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

For which values $\alpha \in \mathbb{R}$ is $f$ integrable? $$f: \mathbb{R}^2 \rightarrow \mathbb{R} : f(x, y) = x \frac{\ln(1 + x^2 + y^2)}{(x^2 + y^2)^\alpha} $$ if $ (x,y) \neq (0, 0) $ and $ ...
0
votes
1answer
75 views

Can I solve this integral with a squared sum in it?

Title says it all. By now I have tried by hand and I think that it is indeed solvable, but I can't handle the very long terms. I tried to run the thing through SAGEs integrator: ...
25
votes
3answers
640 views

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
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4answers
82 views
1
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2answers
28 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
6
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4answers
170 views

Finding $\int\frac{1}{x^{11}+4x^6}dx$

I wanted to find out if there is an easy way to evaluate $$\displaystyle\int\frac{1}{x^{11}+4x^6}dx.$$ I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find ...
2
votes
1answer
18 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...
1
vote
2answers
87 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
4
votes
5answers
92 views

How do I integrate$ \int\frac{1}{e^{2x}+e^x} \,dx $ [on hold]

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
6
votes
2answers
139 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
2
votes
1answer
61 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
0
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3answers
93 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$? [closed]

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
6
votes
3answers
115 views

How do we integrate $(e^x + 2x)^2$?

I need to integrate $$\int(e^x+2x)^2dx.$$ I tried breaking it into $$\int(e^x+2x)(e^x+2x)dx$$ and then integrating by parts, but got stuck at $$ \int (e^x + 2x)^2\,dx = ...
1
vote
1answer
330 views

For each of the following integrals find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral.

$$ \int x\sqrt{7x^2+42x+59} dx $$ There were never any examples quite like this in class, so I'm clueless as to how to figure out which trig function to use.
2
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1answer
56 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
1
vote
2answers
79 views

Help in solving an integral.

I am trying to evaluate this integral, but could not find a solution. I tried it, assuming it to be product of two exponential and then tried integration by parts but it does not lead to anywhere. Can ...
-1
votes
5answers
70 views

Integration of $\displaystyle \frac{e^x-1}{e^x+1}$ w.r.t. $x$ [on hold]

I tried a lot but unable to find out the solution of $$ \int\left(\frac{e^x-1}{e^x+1}\right)dx$$ Please solve it.
2
votes
2answers
110 views

Evaluation of $ \int_{-\infty}^{\infty}\arctan (\frac 1{2x^2})\ \mathrm dx$

Evaluate $$\int_{-\infty}^{\infty}\arctan\left(\frac{1}{2x^2}\right)\mathrm dx$$ And how can I solve it using $$\sum^{\infty}_{x=-\infty}\arctan\left(\frac{1}{2x^2}\right)\quad\text{ and ...
0
votes
1answer
120 views

Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$

How can I prove that $$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$ I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
1
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0answers
20 views

Cauchy formula for repeated Lebesgue integration

Is there an equivalent of the Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for the following \begin{equation} f^{(-n)}(x) = \int_a^x ...
1
vote
0answers
282 views

Inverse Laplace Transform by Complex Inversion Theorem

The questions asks to find $F(t) = L^{-1}\{s^{-1/2}e^{-1/s}\}$ I've made a branch cut along the negative real axis and produced a contour similar to this: Integration along segments BE, LA go to 0 ...
5
votes
4answers
51k views

Definite Integral of square root of polynomial

I need to learn how to find the definite integral of the square root of a polynomial such as: $$\sqrt{36x + 1}$$ or $$\sqrt{2x^2 + 3x + 7} $$ EDIT: It's not guaranteed to be of the same form. ...
2
votes
1answer
28 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
4
votes
3answers
91 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
1
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1answer
35 views

Evaluate the integral $\int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$

I am attempting to evaluate the integral (where $t \rightarrow \infty$) $$I(t) = \int_0^{\infty} e^{\frac{-t(s-1)^2}{2}} \left( \frac{t(s-1)^3}{3} \right) ds$$ which occurs in the calculation of the ...
1
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2answers
35 views

“Trig Substitutions”, I tried half- angle and trig indentity in this one, but doesn't work

I´m really lost in this one. $\int \sin^3 (2x) \cos^2 (2x) dx$ I know that the answer is: $\frac{1}{10}cos^5(2x)-\frac{1}{6}cos^3(2x) + c$ Please help
1
vote
3answers
23 views

integration by parts of $25\, (1-\sin^{2}x)$

I need help solving this integration of parts problem. I've tried a few different solutions and keep getting the wrong answer. This question is in regards to this problem take the integral by parts ...
4
votes
1answer
41 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
0
votes
2answers
421 views

Center of mass of one octant of a non-homogenous sphere

Find the center of mass of that part of the sphere $x^2+y^2+z^2 \le a^2$ having $x,y,z \ge 0$ (that is, the part in the first octant) With density given by $\rho(x,y,z)=(x^2+y^2+z^2)^{3/2}$ It ...
25
votes
1answer
643 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
3
votes
2answers
44 views

Applying the definition of Lebesgue Integral to specific functions

I am fairly sure this question will sound rather naive, but I do have a problem with applying the Lebesgue Integral. Actually this question can be divide in two sub-question, related to two examples I ...
6
votes
3answers
216 views

Finding the definite integral of a trigonometric expression

Find the integral of $$ \int_0^{\frac{\pi}{2}}{{\sqrt{\sin(2\theta)}} \cdot \sin(\theta)d\theta}$$ I got $$I=\int_0^\frac{\pi}{4}{\sqrt{\sin(2\theta)} \cdot (\sin(\theta)+\cos(\theta))d\theta}$$ But, ...
3
votes
2answers
84 views

Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important?

The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ ...
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0answers
35 views

How to evaluate this definite integral? [on hold]

How to evaluate the integral $$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$ Here $a$ and $b$ are some positive real numbers smaller than $1$.
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votes
1answer
52 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!
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3answers
53 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
0
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1answer
20 views

Calculate the flux through a closed surface

While studying for a test I have encountered such a task: Calculate the flux through a closed surface, where $S$ is a boundary of area $V$ with an outward orientation. The data: ...
0
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0answers
22 views

Double integral over a triangle

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function (derivable, integrable over all of $\mathbb{R}^2$). Let $T$ be a triangle in $\mathbb{R}^2$, defined by its vertices : $A=(x_a,y_a)$, ...
0
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0answers
21 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
votes
3answers
37 views

fundamental theorem of calculus 2 [on hold]

Differentiate the following equation with respect to $x$: $$8 + \int_a^x \frac{f(t)}{t^2}\, dt = 2 x^{1/2}$$ Hence, find a function $f(x)$ and real number $a$ such that the above equation is true ...
0
votes
0answers
33 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
0
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0answers
16 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
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0answers
31 views
3
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0answers
17 views

Correct bounds for simple triple integral in rectangular coordinates?

This is homework, so I am not after a solution to this problem. I am required to evaluate the integral $\iiint_{V}y\;dV$. $V$ here is the solid bounded above by the plane $x+y+z=1$ and by the ...
1
vote
1answer
44 views

what is difference between summation and integration? explain with example. [on hold]

I want to distinguish between obtaining process of integration and summation.I.e what we exactly do when we take summation or integration of any function.
4
votes
2answers
164 views

Unnecessary condition of Lebesgue's monotone convergence theorem?

Rudin RCA p.21 Let $(X,\Sigma,\mu)$ be a measure space and $\{f_n\}$ be a sequence of measurable functions on $X$ and suppose that (a) $0\leq f_1\leq f_2\leq\cdots\leq\infty$ for every $x\in X,$ ...
4
votes
1answer
167 views

Calculate the Gauss integral without squaring it first

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a $2-$dimensional integral in the plane and integrate it in ...
0
votes
1answer
45 views

How do we calculate the upper sum and lower sum of an Integral?

How do we calculate the Upper and Lower Sum of an Integral? I am trying to calculate it to for : $$\int_1^2 (3-4x) dx$$ Is there a Formula?