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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36 views

Partial Fraction Decomposition?

For repeated, linear factors, (the term $(x-r)^m$ appears in the denominators for some integer $r$ and $m$, where $m > 1$) there is a partial fraction for each power upto and including $m$. Why? ...
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1answer
62 views

$\Gamma\in C^\infty(0,\infty)$.

How does one demonstrate that the Euler $\Gamma$-function $\Gamma:(0,\infty)\rightarrow\mathbf{R}$ is of class $C^\infty$, taking the integral to be an improper Riemann-integral? I only know how to ...
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35 views

Definite Integral Probabilities

So I have a table of f(x) possibilities like so: x => prob .1 => .3 .2 => .15 .3 => .2 .4 => .15 .5 => .2 The table relates to a SINGLE event ...
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1answer
36 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
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100 views

Numerical integration of Newton-Cotes formulas to calculate coefficients of Adams-Bashforth method

I am having problems with integration of a polynomial - Newton-cortes formula that I plan to use in predictor-corrector integration method (Adams formulas). $$ \beta_{qi} = \int_0^1\prod_{\substack{l ...
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113 views

Calculate $\iint_D x\ln(xy) dx\,dy \text{ where } x = 1, x = e, y = \frac{2}{x}, y = \frac{1}{x}$

Ok this is a sample exercise from the book that I don't know how to solve. Calculate $\iint_D x\ln(xy) dx\,dy \text{ where } 1 \le x \le e , \frac{2}{x} \le y \le \frac{1}{x}$ The answer is ...
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0answers
45 views

Prove that $P_n(1-x)=(-1)^n P_n(x)$ if $n \geq 1$ P is bernoulli polynomials

A sequence of polynomials is defined inductively as follows: $P_0(x)=1; P'_n(x)=nP_{n-1}(x)$ and $ \int^1_0 P_n(x)dx=0$ if $n \geq 1$ (f) Prove that $P_n(1-x)=(-1)^n P_n(x)$ if $n \geq 1$ I ...
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0answers
87 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
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1answer
55 views

Tricky double integral split into two parts

I just cannot get my head around why the two parts of the split integral (5.258) here are equal, or the explanation showing it: ...
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2answers
58 views

How to find this limit using integration?

What is the value of $$\lim_{n \to \infty}\frac{(\sum_{k=1}^{n} k^2 )*(\sum_{k=1}^{n} k^3 )}{(\sum_{k=1}^{n} k^6)}$$ I just know that it has to be done by converting it into an integral. I have no ...
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4answers
138 views

How find this integral $F(y)=\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)(1+(x+y)^2)}$

Find this integral $$F(y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x+y)^2)}$$ my try: since $$F(-y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x-y)^2)}$$ let $x=-u$,then ...
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2answers
107 views

Is This a Bessel Function?

Is the function $$y(x) = c \int_{-1}^1 \cos(xt)(1-t^2)^{n-\tfrac{1}{2}}dt = c \sum \tfrac{(-1)^m x^{2m}}{(2m)!} \int_{-1}^1 t^{2m}(1-t^2)^{n-\tfrac{1}{2}}dt$$ given here a bessel function? It ...
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1answer
44 views

Integration and proof

Use the fact that $f(x)<g(x)$ implies that $\int_a^b f(x)\, dx < \int_a^b g(x) \, dx$ to prove that: $$e-1 \leq \int_0^1\sqrt{(1+x)} e^x \, dx \leq \sqrt(2) (e-1)$$ Please help with this ...
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47 views

Finding work via Line Integrals

The position of an object with mass $m$ at time is $r(t) = at^2 \vec{i} + bt^3 \vec{j}$, where $0 \leq t \leq 1$. Part a asks for the force, which I found to be $2ma \vec{i} + 6mbt \vec{j}$, which is ...
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1answer
47 views

$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ integration

$\int \frac{e^x}{\sqrt{2e^x+e^{2x}}}dx$ I have no idea how to do it:( What is more I don't know how to start it... Any ideas? I will be pretty grateful. I tried some tricks but no one works.
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2answers
99 views

How to prove that $\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1$?

I have some trouble to prove that $$\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1\ ? $$
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3answers
153 views

Integration by parts.

How can I integrate $ \int_{3}^8 \ln \sqrt{x+1}\ dx$ by parts ? Is this step right ? $ \int_{3}^8 \frac{1}{2}\ln(x+1)\ dx $ = $ \frac{1}{2} \int_{3}^8\ln(x+1)\ dx$ $f^{'}(x) = 1 , f(x) = x , ...
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1answer
32 views

How to find $\int_{S^2}f \cdot n \ \text{d}S$ if $f(x,y,z):=(x^3,y^3,z^3)^T$

With $\mathbb{S}^2$ being the unit sphere, how to find $$\int\limits_{\mathbb{S}^2} \vec{f} \cdot \vec{n} \ \text{d}S$$ if $\vec{f}(x,y,z):=(x^3,y^3,z^3)^T$? Apparently, we need to use Gauss. ...
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4answers
129 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
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1answer
113 views

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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2answers
61 views

Partial fractions: why does $\int dt \implies t + C$

I am working on a partial fraction problem here, I understand everything in the problem except $t+C$, so I'd like to know where did the $t+C$ come from ? I want to solve this integral $$ \int ...
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0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
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1answer
61 views

Evaluating $\int\frac{3x+1}{2x^2-2x+3}dx$

Sorry I don't know how to use MathJaX but i've got a problem here that nobody seems to be able to explain to me. $$\int{3x + 1 \over 2x^{2} - 2x + 3}\,{\rm d}x.$$ It seems rather simple at first ...
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2answers
65 views

General solution to the integral of 4/x

A friend asked me what was the solution to the problem (which was on his test)$$\int\frac4xdx$$ I proceeded to tell him that you can take out the 4 in the numerator, and then just take the integral of ...
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1answer
34 views

Invariance of the Haar measure — upon inverses?

To simplify, let's assume $\mu$ is an invariant Haar measure on a commutative locally compact group $G$. Then, this means that $\mu$ is invariant under translation $\mu(U)=\mu(aU)$. However, I ...
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2answers
64 views

Constant area under $e^{-Ax^2}$ for different $A$

I am trying to find a solution to calculate relationship between an amplitude and boundaries of a Gaussian function so that an area under the curve is constant, let's say 2. I found a solution via ...
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3answers
205 views

A tricky Definite Integral

What is the value of $$\int_{\pi/4}^{3\pi/4}\frac{1}{1+\sin x}\operatorname{d}x\quad ?$$ The book from which I have seen this has treated it as a problem of indefinite integral and then directly put ...
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39 views

Question about finding volume using integration?

The question is: Find the volume of the solid whose base is a circle $x^2 + y^2 = 81$ and the cross sections perpendicular to the $x-axis$ are triangles whose height and base are equal. Now what the ...
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3answers
180 views

solving $\int x^7\sqrt{3+2x^4}dx$

I'm trying to solve $\int x^7\sqrt{3+2x^4}dx$ All I have so far is Let $u$ = $3+2x^4$ $du$ = $8x^3$ $dx$ $\frac{du}{8x^3}$ = $dx$ Therefore, $\int x^7\sqrt{u}$ $\frac{du}{8x^3}$ ...
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1answer
29 views

Appreciate help with solving a probability density function for its constant term

I am using StackOverflow a lot for asking and answering programming related questions, and I hope it is appropriate if I'd ask my question below on here on this sister-site. If not, please let me know ...
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1answer
27 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
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55 views

Real Analysis Integration Question

Still working through some things that I don't quite understand. I think this will make considerably more sense once I'm actually enrolled in the course this summer. For now, self study it is... ...
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2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
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38 views

Need help evaluating $\lim\limits_{n \to \infty} \frac{1}{n} \int_1^n \Vert\frac{n}{x}\Vert dx$

$$ \mbox{Evaluate}\quad \lim_{n \to \infty}{1 \over n}\int_{1}^{n}\left\Vert\,n \over x\,\right\Vert \,{\rm d}x $$ Where $\left\vert\left\vert\, x\,\right\vert\right\vert : \mathbb{R} \to \mathbb{R}$ ...
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1answer
50 views

Multivariable Calculus Green's Theorem Along Two Separate Curves

I have a curve which starts on $(-2,0)$ then goes to $(2,0)$ along the curve $y=4-x^2$ then back to the point $(-2,0)$ along the curve $y=x^2-4$. I have to compute the line integral $$\int -4x^2y \ ...
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1answer
134 views

evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$

I'm trying to evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$ This is what I got so far: Let $u$ = $1-2x$ $x$ = $\frac{u-1}{-2}$ $du$ = $-2$ $dx$ $\frac{-du}{2}$ = $dx$ Therefore, ...
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2answers
89 views

Integral of $x^2e^{-ax^2}$

Hey guys I need to find the following integral using integration by parts and not the gamma function. Also there is an a constant a in the exponential function. So it is actually $x^2e^{-ax^2}$. ...
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1answer
46 views

Question about finding the volume of a Sphere to a certain point

I've done a few things but I cant seem to figure out how to solve this. Any help please?
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5answers
211 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
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2answers
80 views

Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$

Hi everyone I have been trying to prove that that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$ . Heres my attempt: LS: $ \int \limits_a^b f(x) dx$ = $ \int \limits f(b) - \int ...
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1answer
57 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
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1answer
59 views

Proof of PDF Integrals

Hi guys my professor gave us some sample proofs to try at home and I was having trouble with 4 of them. I figured out how to do part (a) by using polar coordinates but cannot wrap my head around the ...
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2answers
113 views

Computing $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx$

I'm trying to compute the following Integral $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx$ This is what i've got so far: $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx =\int_{0}^{\pi} {\cos(x)2\sin(x)\cos(x)}dx = ...
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1answer
57 views

Evaluating integral by parts.

Evaluate the following integral. $$ \int e^{2x}\sin{5x}\ dx $$ What I have tried : $ g(x) = \sin5x , f^{'}(x) = e^{2x} , f(x) = e^{2x} $ $$ \int e^{2x}\sin{5x}\ dx = e^{2x}\sin{5x} -\int e^{2x}\ ...
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2answers
49 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
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1answer
48 views

Evaluating an integral by parts.

Evaluate the following integral. $$ \int x^2 e^x\ dx $$ What i have tried : $ f^{'}(x) = e^x , f(x) = e^x , g(x) = x^2$ $$ \int x^2 e^x\ dx = e^x\ x^2 - \int e^x\ 2x\ dx $$ $ f^{'}(x) = e^x , ...
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0answers
86 views

How to show a curve has a bertrand mate? (differential geometry)

Suppose $Y$ is a $C^2-$ arc-length parametrized curve on the unit sphere. For any nonzero constant $\lambda$ and $0 <\theta< \frac{\pi}{2}$,define: $α(t)= \lambda (∫Y(s)ds+ ...
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1answer
33 views

I have some approximate integral calculation. Is there a clean way to prove it?

Let: $P(R)=\int_R^{\infty}F(z)e^{-z}dz$ where $F(z)$ is the CDF of some discreate positive R.V. denote by $U$. Integrate by parts: $P(R)=(-F(z)e^{-z})_R^{\infty}+\int_R^{\infty}F'(z)e^{-z}dz$ The ...
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2answers
122 views

Inverse Functions and $u$-Substitution

Back in my undergrad days I wrote a false proof of the following. Problem. Prove that $\displaystyle\int_0^{2\pi}\frac{dx}{1+e^{\sin{x}}}=\pi$ Proof. Integrating by parts gives $$ ...
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2answers
62 views

Question about apply integrals in finding volume of a pyramid?

The answers entered already is what I got but either one or both are wrong. Can someone help me solve this problem?