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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3
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2answers
46 views

How to calculate $f'(0)$ and $f'$ $(\sqrt{2})$ while $f(x)$= $\int_{x}^{x^3} e^{t^{2}}dt\ $?

How to calculate $f'(0)$ and $f'$ $(\sqrt{2})$ while $f(x)$= $\int_{x}^{x^3} e^{t^{2}}dt\ $? I thought about using the fundamental theorem of calculus, but im not sure im fully aware of how to use ...
3
votes
1answer
45 views

Riemann-Stieltjes Integrability and Convergent Series

Let $\alpha_{n=1}^{\infty}$ be a sequence of monotonically increasing functions on $[a.b]$ such that the series $\sum_{n=1}^{\infty}\alpha_{n}(a)$ and $\sum_{n=1}^{\infty}\alpha_{n}(b)$ converge. ...
3
votes
2answers
91 views

Trigonometric integrals

How do I evaluate this indefinite integral ? Integral $$\int\frac{x^2+n(n-1)}{(x\sin(x)+n\cos(x))^2}dx$$ What type of integral is it ? Is there any intuition involved in the approach to solve it? ...
0
votes
1answer
40 views

Evaluation of Spence's function.

Spence's function is defined as $${\rm Li}_2 (z)=- \int_0^z \frac{\ln(1-u)}{u} \, du $$ where $$z \in {\mathbb C} \setminus [1, \infty )$$ For $|z|<1 $ $${\rm Li}_2 (z)= \sum_1^ \infty \frac{ ...
5
votes
2answers
111 views

evaluate $\int \frac{\tan x}{x^2+1}\:dx$

$$\int \frac{\tan x}{x^2+1}\:dx$$ I used By-parts method setting $u=\tan x$ and $dv=\frac{1}{x^2+1}dx$, but then I got an integral that's more complicated I also thought of trigonometric ...
4
votes
1answer
92 views

Integral of a square compared to the square of an integral

What can be said about a complex valued, continuous function $f$, defined on $[0,1]$, such that: $$ \int_0^1{|f|^2}=\left|\int_0^1{f}\right|^2 $$ I encountered this form as part of an exercise. ...
0
votes
1answer
35 views

Continuous Annuity Question

I need to calculate the present value of a level continuous annuity which pays $1000/mo. for 10 years. The force of interest is 5/(3+2t). I tried taking the integral of e^(integral of force of ...
1
vote
1answer
44 views

Verifying Stokes' Theorem on two intersecting cylinders

Let $S\subset \mathbb{R}^3$ be the portion of the cylinder $y^2 + z^2 = 4$ with $z>0$ and $x^2 + y^2 ≤ 1$. Let $\mathbf{f}(x,y,z) = (zx-y)\mathbf{i}$, where $\mathbf{i}$ is the usual unit vector ...
1
vote
0answers
21 views

Consider the equation: $x' = f(t,x)$. Prove that there is a two-way correspondence between the initial and the limits of the solutions.

Consider the equation: $$x' = f(t,x)$$ wherein, $$|f(t,x)| \leq \phi(t)x, \forall(t,x) \in \mathbb{R}\times \mathbb{R} $$ $$ \int^{\infty}\phi/(t)< \infty $$ If in addition, $f$ satisfying: $$ ...
3
votes
3answers
353 views

Given an integral equation, integrate the function.

$$f(x)=x+\int_0^1t(x+t)f(t){\rm d}t$$ Then what is $$\eta=\int_0^1f(x){\rm d}x$$ Ok you can write: $$\eta=\int_0^1\left(x+\int_0^1t(x+t)f(t)dt\right){\rm d}x=\frac12+\int_0^1\int_0^1t(x+t)f(t){\rm ...
7
votes
1answer
388 views

Integrating a rational function.

How to integrate $$\int_1^{\infty}\frac{2x^3-1}{x^6+2x^3+\sqrt3x^2+1}{\rm d}x$$ The bottom is not factorizable hence no partial fractions. There seems no other way.
0
votes
2answers
44 views

Show $\int_{-\infty}^{\infty} e^{-it^2}dt=(1-i)(\frac{\pi}{2})^{0.5}?$

Show $\int_{-\infty}^{\infty} e^{-it^2}dt=(1-i)(\frac{\pi}{2})^{0.5}?$ This is in my formula sheet but I'm intrigued as to how it is proved?
0
votes
0answers
22 views

Increase of the rms during diffusion - Derivatives calculation

We consider a diffusion equation of the form $$ \frac{\partial F}{\partial t} = \frac{\partial }{\partial x} \!\cdot\! \left[ - \mathcal{F} (t , x) \right] \, $$ where ${ \mathcal{F} (t ,x) }$ is the ...
3
votes
0answers
62 views

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
0
votes
3answers
58 views

Given several integrals calculate ${\int\limits_5^6}$ f(x) $dx$

Let $\int\limits_4^7 f(x)\,dx = 2$, $\int\limits_6^7 f(x)\,dx = 17$, and $\int\limits_4^5 f(x)\,dx = 3$ Calculate $$\int\limits_5^6 f(x)\,dx$$ I guess I am to assume that $f(x)$ is the same in all ...
1
vote
2answers
36 views

convert riemann sum $\lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n}$ to integral notation

The limit $ \quad\quad \displaystyle \lim_{n\to\infty}\sum_{i=1}^{n} \frac{15 \cdot \frac{3 i}{n} - 24}{n} $ is the limit of a Riemann sum for a certain definite integral $ \quad\quad ...
1
vote
1answer
63 views

how to compute the integral $\int_0^1 (1-x^p)^n dx$?

For constants $n$ and $p$, how to compute the integral $\int_0^1 (1-x^p)^n dx$ ? I saw a solution using hypergeometric function and another using incomplete beta function here: ...
1
vote
1answer
27 views

Finding $\frac{d}{dy} \int_{a}^{y} f(x,y)dx$ when $\int f(x,y)dx$ is non-trivial

I ran into a problem where I had to find the following $$\frac{d}{dy} \int_0^y \sqrt{x^4+(y-y^2)^2}dx $$ and was at a complete and utter loss as to where to begin. Any and all insights regarding ...
2
votes
0answers
48 views

Is this integral is right or wrong?

We did this exercise in class in a way, but at home I tried to solve it in a different way and I do not know if it is right or wrong. May you help me please? $\mathbf{\int tan^{5}x \, \, \, sec^{4}x ...
0
votes
0answers
37 views

Integrate $sin(ax)/(x(x^2 + b^2)^2)$ from 0 to infinity, where a and b > 0. [closed]

Integrate the following function, where $a,b > 0$. $$\int_0^\infty\frac{\sin(ax)}{x(x^2+b^2)}\mathrm{dx}$$
0
votes
2answers
51 views

How to evaluate $\int{\sqrt{tan(x)}dx}$ [duplicate]

This might be a simple question, but how do you evaluate $\int{\sqrt{tan(x)}dx}$ ? I've tried substitution with $u=tan(x)$, that introduced a $sec^2(x)$, I've also tried integration by parts, still ...
1
vote
3answers
50 views

How to calculate $\frac {d}{dx}$ ($\int_{1}^{x^2} \sqrt{ln(t)}\,dt$), when $|x|>1$

How do I calculate $\frac {d}{dx}$ ($\int_{1}^{x^2} \sqrt{ln(t)}\,dt$), when $|x|>1$ ? After thinking about this I concluded that the operation $\frac {d}{dx}$ gets us back the original function. ...
1
vote
1answer
36 views

Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'?

For me, intuitively, integral $2\pi y~dx$ make more sense. I know intuition can not be proof, but by far, most part of math I've learned does match with my intuition. So, I think this one should 'make ...
0
votes
1answer
26 views

Finding $y=b$ that dissects the area between $ y=36, y=12, y=x^2$ into 2 equal halves.

Finding $y=b$ that dissects the area between $ y=36, y=12, y=x^2$ . what I did is solving the following equation: $\int_{0}^{6} 36-x^2\,dx$ - $\int_{0}^{\sqrt{b}} b-x^2\,dx$ = $\int_{0}^{\sqrt{b}} ...
6
votes
2answers
133 views

Prove that $\int^2_0 x(8-x^3)^\frac{1}{3}dx=\frac{16\pi}{9\sqrt{3}}$

I have to prove that: $$\int^2_0 x(8-x^3)^\frac{1}{3}dx=\frac{16\,\pi}{9\sqrt{3}}.$$ I tried substituting $x^3=8u$ but I just got stuck. Any help would be appreciated.
0
votes
1answer
29 views

Showing that a function is L1

I have been struggling with this problem; it should just use some basic inequalities, but having difficulty getting them in the right order. Let $f \in L^2(\mathbb{R})$ such that it is also the case ...
4
votes
2answers
274 views

How is this definite integral solved: $\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $?

$$\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $$ Tried partially integrating, had no luck.. Any thoughts?
1
vote
0answers
30 views

Evaluate $\int\cot(\beta x)\sin(\alpha x) \text{d}x$. [closed]

Evaluate $$\int\cot(\beta x)\sin(\alpha x) \text{d}x$$ for non zero $\alpha, \beta\in\mathbb R$ and $\alpha\neq\beta$.
3
votes
4answers
99 views

Showing that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges (Baby Rudin Exercise 6.9)

Motivated by Baby Rudin Exercise 6.9 I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges. My attempt: $\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty ...
5
votes
4answers
62 views

Finding the integral $\int_0^1 \frac{x^a - 1}{\log x} dx$

How to do the following integral: $$\int_{0}^1 \dfrac{x^a-1}{\log(x)}dx$$ where $a \geq 0$? I was asked this question by a friend, and couldn't think of any substitution that works. Plugging in ...
1
vote
2answers
42 views

How can we use Fubini's theorem to simplify $\int_0^r\frac 1{\sigma^{n-1}}\int_0^\sigma\rho^{n-1}f(\rho)\;d\rho\;d\sigma$?

Let $f:[0,\infty)\to\mathbb{R}$ and $R>0$. How does Fubini's theorem imply $$\int_0^r\frac ...
2
votes
2answers
35 views

Solving the indefinite integral of a trig function

I'd like to ask for some feedback on my calculation. Please let me know if you spot any mistakes in my technique: $$\int{\frac{1}{\sqrt{x}}\sin^{-1}{\sqrt{x}}}\,\,dx$$ Using substitution: $$u = ...
1
vote
1answer
30 views

$\int_1^n g(x)dx = \log (n!) - \frac {\log n}{2} - \frac{1}{8n}+ \frac{1}{8}$

Let $m \in \mathbb N$ and $g(x)=\frac{x}{m}-1+\log m$, if $m-\frac{1}{2} \le x<m+\frac{1}{2}$ I need to prove that $\int_1^n g(x)dx = \log (n!) - \frac {\log n}{2} - \frac{1}{8n}+ \frac{1}{8}$. I ...
0
votes
1answer
21 views

Solving the definite integral of trig function

I'd like to ask for some feedback on my calculation. Please let me know if you think it's correct, or if I messed up somewhere: ...
2
votes
2answers
51 views

Variables of integration.

(originally a picture from a textbook:) $\displaystyle f(x) = \int_1^x \frac {\ln t}{1 + t} \, \mathrm dt \text{ for } x > 0$ Now, $\displaystyle f(1/x) = \int_1^{1/x} \frac {\ln t}{1 + ...
0
votes
0answers
18 views

Integrability of a function, $(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$.

Is the function $$(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$$ integrable wrt. the lebesgue measure on $(\mathbb{R}^{2},\mathbb{B}_{2})$? I have shown that it's not integrable, ...
0
votes
2answers
38 views

Basic question about integrating by parts

Suppose I want to solve the following: $\int \arcsin(t) \space dt=?$ In order to solve this I would use integration by parts: $\int \ uv'dx=uv-\int \ u'v \space dx$ If I let $v'=\arcsin(t)$ then ...
2
votes
4answers
57 views

Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
3
votes
3answers
48 views

Is there another way to solve $\int \frac{x}{\sqrt{2x-1}}dx$?

$$\int \frac{x}{\sqrt{2x-1}}dx$$ Let $u=2x-1$ $du=2dx$ $$=\frac{1}{2}\int \frac{u+1}{2\sqrt{u}}du$$ $$=\frac{1}{2}\int (\frac{\sqrt{u}}{2}+\frac{1}{2\sqrt{u}})du$$ $$=\frac{1}{4}\int ...
3
votes
1answer
751 views

A difficult integral (expectation of the function of a random variable)

For $H>L$ , $p,q,\alpha,\beta>0$, and B(.,.) the beta functon, trying to solve this integral: $$\mathbb{E}(X)=\frac{\alpha H }{\beta B(p,q)}\int_0^H \frac{x \left(\frac{-H \log ...
5
votes
1answer
56 views

Compute $\iiint_V \sin^2 (x + y + z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is an ellipsoid.

By performing a suitable scaling and rotation of the coordinates, or otherwise, evaluate $\iiint_V \sin^2{(x + y + z)}\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}x$ where $V$ is the region ...
1
vote
0answers
50 views

Is this integral function bounded?

Let $n \in\mathbb{N},n\geq2$, $\gamma\in\mathbb{R},\gamma<n-1$. Let $\Omega$ be a open and bounded subset of $\mathbb{R}^n$ regular enough ($C^2$ I think). It is known that ...
5
votes
4answers
128 views

Difficult improper integral: $\int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$

How can I find a closed-form expression for the following improper integral in a slick way? $$\mathcal{I}= \int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$$
3
votes
1answer
31 views

Definite integral of exponential function

I'd like to get some feedback on an integral calculation, if anyone might spot something wrong with my work: $$\int_4^9{\frac{2e^{\sqrt{x}}}{\sqrt{x}}}\,\,dx$$ Using substitution, let $$u = ...
1
vote
2answers
48 views

Troubles with this $\int{\sqrt{x^2-2x-1}}dx$

I was practising integrating and I have problem with this one: $$\int{\sqrt{x^2-2x-1}}dx$$ I wanted to do it with substitution. Let $$t=\sqrt{x^2-2x-1}$$ Then $$x=1\pm\sqrt{t^2+2}\quad ...
7
votes
0answers
87 views

Find the closed form of $\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$

I was thinking of the generalization of the problem here, that is $$\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$$ Maybe you recommend me some tools? ...
1
vote
1answer
49 views

Calculate $\int_0^{2\pi} \frac{sin(t) + 4}{cos(t) + \frac{5}{3}}dt$

I have to calculate $\int_0^{2\pi} \frac{sin(t) + 4}{cos(t) + \frac{5}{3}}dt$ using complex analysis. I was thinking of setting $z(t) = re^{it} $ but I'm not sure what $r$ to pick or can I just pick ...
0
votes
1answer
47 views

How do I integrate this

How do I perform this integration? $$\int_{0}^b x\frac{2b(b^2-x^2)}{(b^2+x^2)^2} dx = \ ?$$ I tried using integration by parts and arrived at an expression $$\int_{0}^{b} \frac {6b^3x^3 - ...
0
votes
0answers
10 views

If $C=M \times (0,1)$, what is an integral over $\partial C$?

Let $M$ be a compact Riemannian manifold. Define $C=M \times (0,1)$ so that $\partial C = M \times \{0\} \cup M \times \{1\}$. If $f:\partial C \to \mathbb{R}$, is then the integral over $\partial C$ ...
6
votes
3answers
198 views

How to solve $\int_0^{\frac{\pi}{2}}\frac{x^2\cdot\log\sin x}{\sin^2 x}dx$ using a very cute way?

Few days ago my friend gave me this integral and i cant get how to solve this. The integral is:$$\int_0^{\large \frac{\pi}{2}}\frac{x^2\cdot\log{{\sin{x}}}}{\sin^2{x}}dx$$