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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4
votes
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 $ ...
1
vote
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 ...
1
vote
4answers
79 views

How does $\int (\cos(x))^{-2}dx$ equal to $\tan(x)$?

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
4
votes
5answers
90 views

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

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
2
votes
1answer
60 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 ...
-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.
1
vote
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 ...
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 ...
1
vote
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
vote
2answers
34 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
4
votes
3answers
90 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
vote
3answers
22 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 ...
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 ...
4
votes
1answer
40 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 ...
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 ...
-1
votes
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 $1$.
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 $L^1$ ...
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, ...
0
votes
1answer
74 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: ...
2
votes
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
votes
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
-8
votes
1answer
50 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!
0
votes
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
votes
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
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
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
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 $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
-1
votes
0answers
31 views

When we take integration of any function, then what exactly we do with it? [on hold]

Ex. $\int 2x\, dx= x^2$Then what we have exactly done with function.
3
votes
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.
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?
-1
votes
0answers
26 views

Demonstrate the convergence of an integral [on hold]

Can anyone demonstrate that the following integral is convergent? $$\int_0^{\tau"}\left(\int_0^{\tau'}\frac {1}{\tau^2|\ln\tau|^p}d\tau\right)d\tau'$$ $p$ is a constant $>12$.
0
votes
0answers
12 views

Evaluating and Simplifying a Double Integral

I have an integral as follows $f(t) = \int_r^\infty \frac{(sP)^{1-\rho}t^{-\alpha/2}}{1+(sP)^{1-\rho}t^{-\alpha/2}} \;dt$ I wish to get rid of the $s$ in $f(t)$ because this is an inner integral ...
0
votes
0answers
15 views

Normal convolved to Exp(polynomial)?

Is there an analytic solution for a Normal (normalized Gaussian) distribution of variance $v$ convolved to $e^{y(x)}$, where $y(x)$ is an $m$-th order polynomial? Assume that $m$ is even and the ...
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 ...
2
votes
3answers
92 views

Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
0
votes
2answers
32 views

How to solve an integral with the use of arcsine

The specific question is the following, $$\int_{-a}^x \sqrt{a^2-x^2}\,dx$$ We are also given that $0\le x\le a$ Thank you very much for helping.
0
votes
0answers
39 views

is it possible to evaluate any definite integral using the definition of the definite integral?

I was evaluating definite integrals using the fundamental theorem, however, out of curiosity, I wanted to see if it was possible to evaluate the following, using the definition of the definite ...
0
votes
0answers
13 views

upper-band of the Integral expression

Consider below integral expression $$\int_{0}^{\infty}g(y)[\int_{a}^{\infty}(1-e^{-(k+y)x})f(x)dx ]dy \ \ \ \ (1)$$ Where, we know: $$f(x)>0\ ,\ \ a\leq x \leq \infty$$ $$\ k>0$$ $$g(y)>0\ ...
0
votes
3answers
29 views

Help with a derivative of integral please.

I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. ...
0
votes
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: ...
5
votes
1answer
86 views

A triple integral dancing in the unit cube

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 ...
5
votes
0answers
40 views

Different representations of Appell hypergeometric series

The (first) Appell series: $$F(a; b_1, b_2; c \mid z_1, z_2) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \sum_{n_1+n_2=n} (b_1)_{n_1} (b_2)_{n_2} \, \frac{z^{n_1}}{n_1!} \frac{z^{n_2}}{n_2!}$$ can be ...
3
votes
0answers
64 views

Evaluate $\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$ [duplicate]

Evaluate $$\int \dfrac{1}{\sqrt{x-1}+\sqrt{x}+\sqrt{x+1}} \ \mathrm{d}x$$ I tried rationalizing the denominator by twice multiplying, but it didn't do any good. I also tried trig ...
2
votes
2answers
80 views

Integral: $\int \sqrt {\sin x} \, \mathrm{d}x$

I want to find $$\int \sqrt {\sin x} \, \mathrm{d}x$$ Now what I think that this can not be integrated without any definite boundary given o.w we can shhift it to gamma function or directly using ...
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
18 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
1
vote
0answers
17 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
2
votes
4answers
54 views

Integral $\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$

I need a hint how to start solving this integral: $$\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$$
0
votes
2answers
50 views

Do nested integrals exist?

I have a problem that involves evaluating (or at lest simplifying) the expression $$\int_{0}^{x}\int_0^{x'}f(y)dy dx'.$$ Playing around with Riemann sums has lead me to believe that this is just ...