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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21 views

Surface area of cylindrical surface using double integrals

Please help lead me in the right direction for this question, I'll give a description of my progress so far. My understanding is that the formula for the surface area is given by this equation: ...
2
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1answer
227 views

Definite integral over triple products of higher order Bessel functions.

As a follow up to this question I am also interested in a symbolic closed form for this integral $$\int_0^\infty d r \,r^2\, j_{n_1}( k_1 r)\, j_{n_2}( k_2 r)\, j_{n_3}( k_3 r)\,, $$ where $j_n(r)$ ...
2
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0answers
24 views

Solving String Vibration Using Integral Transform

$$U_{tt} - c^2 U_{xx}= -g$$ where BC: $U_{x}(0,t)=a\sin(ωt)$ IC: $U(x,0)=0$, $U_{t}(x,0)=0$ where $c, g, A$ and $ω$ are positive constants Normally I wouldn't post for help here but I am ...
3
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1answer
43 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
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2answers
47 views

Solve $\int\ 3x\cos(2x)dx$ with integration by parts

I have this problem here that is asking for me to use Integration by Parts. I solved it out but it seems as if it can be an ongoing function. I was told you can work with it if the second time you ...
3
votes
3answers
77 views

Solve $\int\ e^x \sin(9x)\,dx$ using integration by parts

I have this Integration by Parts question that I can't seem to find an answer to. The question is: $$\int\ e^x \sin(9x)\,dx$$ I used u-substitution: $$u=e^x,du=e^x\,dx$$ $$dv=\sin(9x)\,dx, ...
0
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1answer
38 views

How to evaluate this infinite sum

I want to find $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx$. From binomial theorem, $(1-x)^{\frac{1}{3}}= \sum_{0}^\infty (-x)^n\binom{\frac{2}{3}}{n}$. Then $\int_0^1(1-x)^{\frac{1}{3}}x^{1/3}dx= ...
3
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1answer
165 views

Example of $f,g: [0,1]\to[0,1]$ and riemann-integrable, but $g\circ f$ is not?

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if ...
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0answers
20 views

Writing an integral for a bounded volume

Good day folks! I'm trying to write an integral which could represent the volume of the shaded area below. However my integral calculus is quite poor, so I need some help on that equation. I ...
1
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2answers
44 views

If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ …

Suppose $f : [a,b] \to \mathbb{R}$ is continuous and $g \in \mathcal{R}[a,b]$ with $g(x) \ge 0$ for all $x \in [a,b]$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, dx = ...
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0answers
25 views

Finding the normal vector of a surface (Flux of a vector field n*dS expression)

This problem is practice for a final exam. Let S be the closed surface whose bottom face B is the unit disc in the $xy$-plane and whose upper surface U is the paraboloid $ z = 1 − x^2 − y^2 , z \geq ...
2
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0answers
25 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
1
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1answer
39 views

Derive a formula for the volume of the wedge in terms of the constants a, b, c.

Derive a formula for the volume of the wedge in terms of the constants a, b, c. Seeing a similar triangle, I see that $\frac{x}{y}=\frac{c}{b}$, $y$ being the distance from the $a$ line to the ...
2
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4answers
159 views

Indefinite Integral $\int \frac{dx}{\sqrt {ax^4-bx^2}}$

I am trying to Integrate $$ I=\int \frac{dx}{\sqrt {ax^4-bx^2}}, \qquad a,b\in \mathbb{R}. $$ Thanks. I tried to do $x=\sin \phi$ $$ \int \frac{\cos \phi\, d\phi}{\sqrt{a\sin^4 \phi-b\sin^2 ...
9
votes
3answers
685 views

Pretty Simple Integral

I am trying to find the following indefinite integral: $$ \int \sqrt{x^2+x^4}dx $$ At first, it seems like an easy u-substitution after we factor out an $x^2$ from the square root, but when we do ...
1
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1answer
44 views

Evaluate $\int_0^1\int_x^1 e^{x/y} dy\,dx$

I need some help to solve the following: $$\int_0^1\int_x^1 e^{x/y} dy\,dx$$ I guess it is related with change of variable, but I can't figure out which one. Thanks in advance. Regards.
1
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1answer
21 views

Volume of a solid formed as vertical limit goes to infinity

Here's the question: The way I have it set up currently is as follows: $V = \pi \lim_{a \to \infty} \int_1^a (a-1)^2 - (\frac{1}{\sqrt{x^5}} - 1)^2$ But how do I go from here? And is the working ...
4
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3answers
103 views

Compute the fourier coefficients, and series for $\log(\sin(x))$

I posted a similar question with a bad response, so I am retrying with hopes of better knowledge. The fourier series is in the form: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + ...
4
votes
1answer
47 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
5
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1answer
62 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
2
votes
5answers
120 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
0
votes
0answers
22 views

Any way to simplify integral of Confluent Hypergeometric Function of the First Kind?

The integral is this: $$\int_{-\log n}^{0}e^{t(1-s)} \cdot z \cdot {}_1F_1(1-z, 2, t) dt $$ Is there a way to write this in terms of special functions that eliminates the integral and doesn't use ...
1
vote
2answers
71 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
9
votes
3answers
129 views

How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$

Evaluate $$\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$$ where $a$ is a real parameter $a\geq1$. I can easily find the definite integral for $a=1$. It is $\sin(1)$. In wolframalpha.com when I put ...
3
votes
3answers
49 views

Using trig substitution to solve for integration?

So I used a trig sub for this problem: $$\int \frac{1}{x^2\sqrt{9-x^2}}dx.$$ ${x=3\sin\theta}$ ${dx=3\cos\theta\ d\theta}$ ${\sqrt{9-x^2}= 3\cos\theta}$ I ended up with $$\frac19 \int \frac{ ...
0
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2answers
35 views

Differential equation problem. Integrating the logistic equation. [duplicate]

I would like to know how to integrate or rather solve this: $$ \frac{dP}{dt} = kP(L-P). $$ I have the solution, but I would like to know how to arrive at it. I have been told it involves separation ...
18
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2answers
992 views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
5
votes
2answers
87 views

Antiderivative of $\frac{1}{\ln(x)}$?

I was looking on wikipedia, and found that the following expression cannot be expressed in terms of elementary functions: $$\int\frac{1}{\ln(x)}\text{d}x$$ Although the function looks simple, why is ...
0
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0answers
15 views

Is it always possible to use Chasles to decompose an integral ?

Well, I'm in "classe préparatoire" and I always learn that if f is a continuous fonction integrable on [a,b] and if c is in [a,b] then with Chasles relation we have : $$ \int_a^b f(x) dx = \int_a^c ...
1
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2answers
74 views

Solving $ \int_{0}^{2\pi} e^{-x} \lvert \sin x\rvert,dx $

Since it involves an absolute value, I assume I need to split it into two cases? For $ 0 \le x \le \pi $ $$ \int_{0}^{\pi} e^{-x} \sin x\,dx $$ and for $ \pi \le x \le 2\pi $ $$ \int_{\pi}^{2\pi} ...
0
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0answers
43 views

How do we calculate naturally the following integral :

Do you know a natural method to calculate the following integrals: $$ I = \int_{\mathbb{P}^{1} (\mathbb{R})} \dfrac{1}{x} dx\quad\text{and}\quad J = \int_{\mathbb{P}^{1} (\mathbb{C})} \dfrac{1}{z} dz. ...
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votes
1answer
20 views

Linear Algebra - Weighted Inner Product of Polynomials [on hold]

Given the weighted inner product $\langle f,g\rangle = \int^1_{-1}f(x)g(x)x^2dx$ How do you find an orthogonal basis of the space $\Bbb P^1$ of polynomials of degree $\le$ 1. And how do you find the ...
1
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1answer
98 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...
4
votes
1answer
170 views

Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
1
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1answer
60 views

Question about the substitution rule. Where did I go wrong?

$$\int (f(x))' dx = f(x) + c$$ if $u=g(x)$ then $$\int (f(u))'du = f(u)+c$$ But $$\int (f(g(x)))'dx = f(g(x))+c$$ Where did I go wrong?
0
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1answer
125 views

Find derivatives of functions with respect to $ x$

Can someone help me with these. Find the derivatives of the following functions with respect to $x$: here $a$ is an arbitrary (fixed) real number. $(a)$ $\displaystyle\int_{a}^{x^4} t^3\ \mathrm dt$ ...
4
votes
1answer
176 views

Evaluation of $I_{a,b} = \int_{1}^{+\infty} \frac{\ \exp{(-at)}}{ 1-bt} \ \mathrm{d}t $

How to evaluate this integral: $$I_{a,b} = \int_{1}^{\infty} \frac{\ \exp\left(-a t\right)}{ 1-b t} \mathrm{d}t $$ where $a, b \in R^*_+$ ?
0
votes
1answer
36 views

Behavior of the following function at $x=0$ singularity

I am trying to do the following integral: \begin{equation} \int\frac{1}{x^{2p}(x-1)^{2q}}\,\mathrm dx \end{equation} for positive $2p$ and $2q$. I want to understand how does this function blow up ...
3
votes
1answer
143 views

Evaluating $\int_0^\infty \frac{\log t}{1+t^2}\,\mathrm dt$ using residues

I want to integrate $$\int_0^\infty \dfrac{\log t}{1+t^2}\,\mathrm dt$$ using the residue theorem. The poles are at $i,-i$. If the integral were from $-\infty$ to $\infty$, I would consider ...
4
votes
3answers
55 views

Is this an identity: $\int_X f \, d\mu=\int_{\mathbb{R}} \mu(f^{-1}(t)) \, dt$?

Let $(X,\mu)$ be a measure space and $f:X \to \mathbb{R}$ an integrable function. Does the following always hold? $$\int_{X}f\,d\mu=\int_{\mathbb{R}}\mu(f^{-1}(t))\,dt$$
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2answers
71 views
1
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1answer
67 views

Integration Question - Not sure how to approach

I have absolutely no idea how to approach this question: $$\int \frac{x^2}{(15+6x-9x^2)^{3/2}} \ \mathrm{d}x$$ I'm almost positive that it has something to do with trigonometric substitution, but ...
17
votes
5answers
251 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
14
votes
7answers
417 views

Evaluation of $\int_0^1 \frac{x^3}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,\mathrm dx$

How does one evaluate the following integral? $$\int_0^1 \frac{x^3}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,\mathrm dx$$ This is a homework problem and I have been evaluating this integral for ...
2
votes
2answers
65 views

Evaluation of $\int\frac{1}{1+(x+1)^{{1}/{n}}}dx$ for $n\in \mathbb{N},$

Evaluate $$\int\frac{1}{1+(x+1)^{{1}/{n}}}\,\mathrm dx$$ for $n\in \mathbb{N}$ $\bf{My\; Try::}$ Let $$(x+1)=t^n\;,$$ Then $$dx = nt^{n-1}dt$$ So $$\displaystyle I = ...
0
votes
0answers
12 views

Proving $\frac{d}{d\theta}\mathbb E\left[ \log\left( \frac{AY+BY+N}{ AY+BY \frac{X}{\theta^{-\alpha}} +N } \right) \right] \leq 0$

Let $X$ and $Y$ be exponentially distributed random variables with means $\theta^{-\alpha}$ and $(1-\theta)^{-\alpha}$, respectively. Simulation results suggest that $$\frac{d}{d\theta}\mathbb ...
11
votes
4answers
269 views

Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$

How do we prove that $$I(m)=\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x=\frac{\log{m}}{m^2-1}$$ I see that $$I(m)=\frac{\partial}{\partial m} \int_{0}^{\pi/2} \arctan({m\tan x}) \ ...
158
votes
5answers
38k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
33
votes
2answers
944 views

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

Regarding this problem, I conjectured that $$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi ...
5
votes
0answers
20 views

Discrete analogue of Green's theorem

Following formula concerning finite differences is in a way a discrete analogue of the fundamental theorem of calculus: $$\sum_{n=a}^b \Delta f(n) = f(b+1) - f(a) $$ We can think about the Green's ...