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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1answer
14 views

Proving an identity using Riemann-Stieltjes Integration?

Prove the following identity using Riemann-Stieltjes Integration: $$\sum_{n=1}^N \frac{1}{n^s} =\frac{1}{N^{s-1}} + s \int_1^N \frac{\lfloor x\rfloor}{x^{s+1}}dx$$ Here's what I have so far: $$ ...
5
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2answers
20 views

Conceptual question on substitution in integration

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = ...
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1answer
30 views

Integral $-\int_0^\infty e^xx\log(1+kx^2)\,dx$

How to Evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
1
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1answer
18 views

Integration by substitution?

$$\int\frac{(\ln x)^{10}}{x}\,dx$$ All I know is that I am supposed to substitute $u=\ln x$. But can someone please explain to me how to find the anti derivative of $(\ln x)^{10}$. I think we are ...
0
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3answers
62 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
2
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0answers
25 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
1
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2answers
32 views

When does a double integral represent a surface area, and when does it represent a volume?

When does $\int_Af(x,y)dA$ represent a surface area geometrically, and when does it represent a volume? In my lecture notes I'm told it represent the volume underneath the surface $z=f(x,y)$, but I've ...
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0answers
26 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
2
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0answers
69 views

Derivative under integral mixed with…

$$f(x,y)=\int_{e^{4y}}^{\ln^3(x)}{\frac{\sin(t)}{t}\,dt}$$ Whats the derivative $\frac{d f}{d t}$, if: $$x(t)=\cos(2+6t).4t^2$$ $$y(t)=\ln(2r+7e^{5t})$$ Really not much to say about this problem ...
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1answer
23 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
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1answer
34 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I've approved (using WolframAlpha) that $\frac{\pi}{2} - x ...
2
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1answer
24 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
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2answers
57 views

How can I prove this integral?

I have to use the identity $b^4-a^4=(b-a)(b^3+b^2a+ba^2+a^3)$ to prove that: $\int_b^ax^3dx=\frac{b^4-a^4}{4}$. I know that you can just do $F(b)-F(a)$ and since the integral of $x^3$ is ...
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2answers
30 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
2
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2answers
19 views

Improper integral calculation - limit at infinity

Will you please help me prove the following limit is zero ? $$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du. $$ Thanks in advance
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2answers
32 views

Evaluating $\int \frac{(1+\cot^2 x)(\cot x)}{\csc x}dx$ [on hold]

Please could someone help with $$\int \frac{(1+\cot^2 x)(\cot x)}{\csc x}dx$$? Thanks
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3answers
22 views

Integration of $\frac{1}{(4x^2-8x+3)^{1/2}}$ [on hold]

Please could someone help with the integration of $\frac{1}{(4x^2-8x+3)^{1/2}}$? Thank you.
1
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1answer
15 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
3
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1answer
21 views

Average distance to a random point in a rectangle from an arbitrary point

I'm interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and ...
2
votes
2answers
51 views

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

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

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
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3answers
27 views

Line integral of two intersecting spheres

How can I find the length of the line formed by two intersecting unit spheres shifted a distance x from each other? Any suggestions to approaching the problem is also greatly appreciate! Thanks!
4
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2answers
25 views

Investigate the convergence of $\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$

Investigate the convergence of $$\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$$ so first of all let's split the integral to: $$I_1 = \int_1^2 \frac{\cos x \ln x}{x\sqrt{x^2-1}}, I_2 = ...
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0answers
25 views

Need help solving this integral

$\int_1^\infty du$$\int_{-2}^2 dv(u-v)e^{-u}$ Do I just evaluate the integrals separately and then multiply the answers together?
2
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0answers
22 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
2
votes
3answers
61 views

Limit evaluation with integral

Evaluate the limit $$\lim_{n\to\infty} \int_0^1 n^2x(1-x^2)^n dx$$ My Proof: We may look at $n$ as a constant and evaluate the integral $\int_0^1 x(1-x^2)^ndx$ (I already moved out the $n^2$). ...
2
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2answers
38 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
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1answer
24 views

Compute integral given 2 other integrals

I want to know which solution is correct. The question states: If f is an integrable function on [1,3], and if $$\int_1^2f(x)dx=4 \space\space\space\space\space\space and \space\space\space\space ...
0
votes
1answer
33 views

What is wrong with my integral solving

Consider the integral $$\int{ \frac{x^2+2x+8}{(x^2-2x)(x^2+4)}}dx$$ I simplify it to $$\int\frac{x^2+2x+8}{x(x-2)(x^2+4)}$$ Then I try to solve it in sum of partial fraction which gives me ...
2
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1answer
28 views

How do we know which variable to substitute in integration by substitution?

Often times, I encountered questions that requires Integration by substitution; however, I am still somewhat confused regarding the choice of values that should be substituted by u since it differs by ...
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1answer
22 views

Can a Function have Multiple Valid Indefinite Integrals

Working with U-substitution, I have to integrate the following. $\int x\cos(x^2)\sin(x^2)dx$ From my understanding you can take the integral by substituting $u$ for either $\cos(x^2)$ or ...
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1answer
30 views

Differentiating both sides with respect to time.

So I have this problem: An active volcanic mountain grows in the shape of a cone while maintaining its base diameter equal to its height. The volume of the mountain increases at a rate of ...
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1answer
33 views

How to compute $\sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda$

Please suggest an efficient method to compute the following integral \begin{equation} I = \sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda \end{equation} ...
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1answer
30 views

Computing a specific line integral

Here is the problem as I have been given it: A curve $C$ is given in Cartesian coordinates by $r(t) = (cos(sin(nt))cost,\; cos(sin(nt))sint,\; sin(sin(nt)))$, with $t$ between $0$ and $2$$\pi$ ...
4
votes
4answers
86 views

Integral $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx$, where $0<c<1$

I am trying to evaluate the following integral: $$\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx,$$ where $0<c<1$. I can't really think of a way to find it so please give me a hint.
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2answers
35 views

Level curves for “unsolvable” integral

Problem: Sketch the level curves of g defined by $$g(x,y)=\int_x^y{e^{-t^2}dt}$$ (The error function does not need to be used here). Attempts at solution: (1) Apparently we could take $y=x$, then ...
2
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1answer
30 views

Integral reducing

I'm not following all the steps to this integral. I understand the exponent manipulation between steps 1 and 2, but I don't understand how the integral goes from step 2 to step 3. (i.e. I don't ...
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2answers
63 views

How to find the integral $\sin^2\sqrt2x$

I need help finding the integral of $\sin^2\sqrt2x$ I started to integrate it using integration by parts: $u=sin^2\sqrt2x$ and $dv=dx$ $\int u \,{\rm d}v = uv - \int v\,{\rm d}u$ But ...
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1answer
28 views

Area of Lemniscate of Bermoulli

I need to find out area of one loop of Lemniscate $r^2 = \sin(2\theta)$. I have tried taking square root and substitution but those haven't led to anything.
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4answers
146 views

Evaluate the sum $x + \frac{x^3}{3} + \frac{x^5}{5} + … $

Evaluate the sum $$x + \frac{x^3}{3} + \frac{x^5}{5} + ... $$ I was able to notice that: $$ \sum_{n=0}^\infty \frac{x^{2n-1}}{2n-1} = \sum_{n=0}^\infty \int x^{2n-2}dx = \lim_{N\to\infty} ...
1
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1answer
39 views

Surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using parametrization

Calculate the surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using a suitable parametrization and the corresponding surface element. I think this set is a ...
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0answers
6 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
0
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1answer
21 views

calculate riemann integral on the triangle [on hold]

Calculate Riemann Integral $$ \int \int_{B} e^{4y^2}dxdy $$ where B is a triangle: (0,0), (0,1), (-1,1) I have no idea what should I do with that integral.
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4answers
38 views

Prove the inequalities without calculating the integrals

$$ \int_{0}^{\frac{\pi}{2}} \sin^4x dx \le \int_{0}^{\frac{\pi}{2}} \sin^3xdx$$ I have tried to define 2 functions $ f, g:[0, \frac{\pi}{2}] \rightarrow \mathbb{R}$ and say that $ f(x) = \sin^4x$ ...
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0answers
7 views

Integrate $(\frac{y}{R})^{3/7}\, dA$

How do I find the integral for: $\displaystyle \bigg(\frac{y}{R}\bigg)^{3/7}\, dA$; where $R =$ pipe radius, $r = $radius from centerline, and $y = R-r$ ? I know I'm supposed to integrate from $y=R$ ...
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0answers
20 views

Calculate the integral of unknown geometric shape [on hold]

This is a 10m long tunnel for walkers. The profile is parabolic. How would you calculate the whole concrete needed for this tunnel? My problem is the geometric shape. :(
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2answers
66 views

$ \int \frac{dx}{4x^2-12x+13}$

This is probably not too hard but i can't get it right: I am trying to calculate $$\displaystyle \int \frac{dx}{4x^2-12x+13}$$. The solution is $\displaystyle ...
1
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3answers
66 views

Use integration by substitution

I'm trying to evaluate integrals using substitution. I had $$\int (x+1)(3x+1)^9 dx$$ My solution: Let $u=3x+1$ then $du/dx=3$ $$u=3x+1 \implies 3x=u-1 \implies x=\frac{1}{3}(u-1) \implies ...
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0answers
26 views

For the following integrals find a and find b [on hold]

In the following picture, what is a=? what is b=?
2
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2answers
46 views

Integrating: $ \;\int \frac{1}{x^2+3x+2} dx $

How can I solve the following integral: $$ \int \frac{1}{x^2+3x+2} dx $$ Should I proceed by changing the variable (substitution)? or should I use integration by parts? Or another method ...