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
46 views

Generalised integral

I want to carry the notion of Riemann integration to a more general setting. I have already given the following axioms on area function defined on an arbitrary Cartesian product: Let $X$ and $Y$ be ...
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1answer
46 views

Derivation of an integration

Can someone explain to me the difference between the results of $ A$ and $B$, where $$A=\frac{d}{dc} \int_{-\infty}^c xf(x) dx $$ $$B= \frac{d}{dc} \int_c^{+\infty} xf(x) dx $$ You can image $f(x)$ ...
0
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0answers
29 views

integration by parts partial derivatives

I have a question about integration by part and application in case with partial derivative. Is the following true ? $$ \int^{+\infty}_{-\infty} f(x,y)g(x,y)dx=f(x,y)G(x,y)-\int^{+\infty}_{-\infty}\...
2
votes
3answers
78 views

Solving an equation involving an integral: $\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$

Determine a pair of number $a$ and $b$ for which $$\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$$ I tried putting $x$ as $1-x$ as the integral wouldn't change but could not move forward from ...
1
vote
1answer
49 views

Iterated Integral with variable substitution

I need to calculate the double integral of the function $f(x,y) = (x+y)^9(x-y)^9$: $\int_0^{1/2} \int_x^{1-x} (x+y)^9(x-y)^9 dydx$ I have a solution but I definitely arrived at it after a sloppy ...
0
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0answers
21 views

When is the incomplete Beta funtion finite?

I am interested in the incomplete Beta function as defined on Wolfram Mathworld, i.e. $$\text{Beta}(z;a,b)=\int_0^z u^{a-1}(1-u)^{b-1}.$$ I can't seem to find any results on the convergence of this ...
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3answers
60 views

Finding Explicit Form of Function Defined by Definite Integral

Let $$f(y) = \int_{-\infty}^{\infty} e^{-x^2} \cos (xy) \> dx$$ One can show that $$f'(y) = - \int_{-\infty}^{\infty}xe^{-x^2} \sin (xy) \> dx$$ I'm interested in making an ODE involving $...
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2answers
36 views

Representing $f(x,y)$ as a Sum of Partial Derivatives

I was attempting an exam question which looked like this: Given the expression: $P(x, y)\text{d}x + Q(x,y)\text{d}y = 0$ Where: $P(x,y) = 6x +9y + 11 \\ Q(x, y) = 9x - 4y +3$ Find a function $f(...
0
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1answer
26 views

Cylindrical Coordinates

In the following example i am looking to find the volume of the solid bounded above by the plane $ z = y$ and below by the paraboloid $ z = x^2 +y^2 $ by the method of cylindridical coordinates. ...
-4
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0answers
52 views

Is this integral equals zero? [duplicate]

I want to calculate this integral $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ How do I calculate this integral?
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1answer
63 views

Find $H'(x)$ if $H(x)=\int_{3}^{x^2} (\sin t)^3 dt$

let $H(x)=\int_{3}^{x^2} (\sin t)^3 dt$. Find $H'(x)$. I understand that this question is related to the fundamental theorem of calculus, but how should I approach it?
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2answers
71 views

Prove: $e^{x^2}$ has an antiderivative above $\mathbb{R}$

The function $e^{x^2}$ is continuous in $\mathbb{R}$ is it sufficient to conclude that it is differentiable in $\mathbb{R}$? If it is, we can define for all $x\in \mathbb{R}$, $A(x)=\int_0^x e^{t^2} \...
4
votes
1answer
144 views

What is relation between these integrals

I know $$ \int_{0}^{\frac{\pi}{2}}\ln(\sin x)dx=-\frac{\pi}{2}\ln(2)$$ What is relation between it and $$\int_{-\infty}^{\ln(4)}\frac{xe^x}{\sqrt{4e^x-e^{2x}}}dx$$ Please guid me. I have sixteen ...
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0answers
41 views

On the vanishing of integrals involving the $\sinh$ function. [on hold]

Suppose for some positive real $\theta$ that $$\int_1^\infty f(x)\sinh(\theta\log \sqrt x) \mathrm{d}x = 0$$ Where $f(x)$ is a non-constant and continuous function of $x$. What necessary properties ...
1
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2answers
35 views

Spherical Coordiantes in vector calc

I need to evaluate this triple integral $$ \int^2_{-2} \int_0^{\sqrt{4-x^2}} \int_0^{\sqrt{4-x^2-y^2}} (x^2+y^2+z^2) \, dz \, dy \, dx $$ My solution: First I identified the solid as being a ...
4
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4answers
62 views

How to solve $\int \frac{1}{x^2+4x+7} dx$?

How to solve $\int \frac{1}{x^2+4x+7} dx$? I think the first step is to write it in the following form: $$\int \frac{1}{(x+2)^2+3} dx$$
0
votes
1answer
21 views

Using cylindrical to find the volume [on hold]

In the following textbook review question I have been asked to find the volume of the solid bounded above by the plane $ z = y $ and below by the paraboloid $ z = x^2 + y^2 $ looking for some help ...
1
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0answers
22 views

Prove: $\omega(f,P)\leq \omega(f,Q)+2nL*\lambda(P)$

Let $P=\{x_{1},...,x_{k}\}$ and $P\subseteq Q$ a refinement of the partition $P$ which is due to adding one point $Q=P\cup\{y\}$, In this case both partition are the same except of on interval $[x_{i-...
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1answer
31 views

Vector calculus triple integral question

Looking to get some help with this triple integration question from my textbook. $\iiint _D 3xydV$ where $D$ is the solid region above the xy-plane bounded below by the cone $ z=\sqrt{x^2+y^2}$ and ...
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0answers
68 views

Mathematics Colleges [on hold]

Can anyone please give me the names of US colleges with the best mathematics program for undergraduates? I came to know about Carneige Mellon university, but it does not offer financial aid to ...
1
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2answers
51 views

Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$ \ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1 $$ I ask this question as I'm ...
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3answers
60 views

How to integrate the following: $\int{\frac{2y'y}{y^2+1}dx}$

I have encountered the following problem: $\int{\frac{2y'y}{y^2+1}dx}$ According to wolfram the solution is: $log(y^2 + 1)$ How was this solution derived and which rules were used?
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1answer
74 views

How to integrate $\int \ln(\cos^2 x) dx$

Integral: $$\int \ln(\cos^2 x) dx$$ I applied parts technique twice, first by expressing it as $-2\int1\cdot\ln(\sec x)$, which is $$I = -2x\ln(\sec x)+2\int x\tan (x)dx$$ Now the second term can ...
6
votes
4answers
2k views

How do you solve the following separable differential equation: y'y = y + 1?

I just started learning about differential equations and encountered following equation: $$ y'y = y +1 $$ Wolfram alpha provided the following explanation: here But I'm not sure how the integration ...
3
votes
1answer
65 views

Proof of the convergence of $\int_0^\infty\sin{(x^4)} dx$ with Riemann-Lebesgue lemma

In this question, a comment from Lucian asserts that the convergence of the integral $$ I=\int_0^\infty\sin{(x^4)} dx $$ is due to the Riemann-Lebesgue lemma. However, I don't immediately see how to ...
0
votes
3answers
64 views

Integral calculation, solving integral issue [on hold]

Can you guys help me solve this integral? $$ \int \frac{x^3}{x^4 + 2x^2 - 6}dx $$
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votes
6answers
127 views

What is the result of $\int_{0}^{\infty} \frac{1}{x(x+1)}\ln(x+1)dx$

Is there a result for $$ \int_{0}^{\infty} \frac{1}{x(x+1)}\ln(x+1)dx $$ If not, is there any upper bound for that? Update: The result is $\frac{\pi^2}{6}$. But how to prove it?
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0answers
42 views

Rigorious formulation of approximation of integral as square for large 2nd derivative.

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
1
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1answer
51 views

Proving the Tautochrone Property

The tautochrone property (meaning equal time) is one of the dynamic properties of an inverted cycloid. This means that if one puts two objects at different positions on a inverted cycloidial shaped ...
1
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0answers
42 views

Taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = ...
2
votes
1answer
28 views

Can I integrate the Lie-algebra (body angular velocity) of a quaternion?

This is my first mathematics question here. So I am trying to model a 3-d rotation rigid body by Euler's equation. Of course quaternion is the place to go. If in each time step I receive the body-...
5
votes
2answers
252 views

Triple Integration in vector calc

NIn the following example I have been asked to find the volume V of the solid bounded by the sphere $ x^2 + y^2 +z^2 = 2 $ and the paraboloid $ x^2 + y^2 = z $ by using triple integration. I am not ...
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1answer
88 views

Is there a closed form expression for integral $\int_{0}^a \frac{1}{x} \ln(x+1) dx$?

Is there a closed form result for integral $$ \int_{0}^a \frac{1}{x} \ln(x+1) dx $$ where $a$ is a positive real value.
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2answers
92 views

Find a constant that minimizes $\int_0^1 |e^x - c| \ dx$

When considering the expression $|e^x - c|$, there are several ways to choose our constant $c$ that minimizes this for different norms. There are standard methods for doing so in the $\infty$-norm and ...
1
vote
1answer
44 views

Work done by a Force Field( Green's Theorem)

Question: Compute the work done by a force Field $ F(x,y)=(2xe^y-x^2y-\frac{y^3}{3},x^2e^y+sin(y)) $ when a particle moves moves around the path describe by $ r(t)=(1+cos(t),sin(t)),0 \leq t \leq \pi ...
13
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1answer
291 views
+50

A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt $

$$ \mathbf{\mbox{Evaluate:}}\qquad \int_{0}^{1} \sqrt{\frac{1}{\left(1 - t^{2}\right)^2} - \frac{\left(n + 1\right)^{2}\,t^{2n}}{\left(\, 1 - t^{2n+2}\,\,\right)^{2}}} \,\,\mathrm{d}t $$ where $n$ ...
0
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0answers
24 views

Line integral with differentials (cylindrical/spherical)

How can I write a line integral of a vector field with exact differentials in cylindrical and spherical coordinates? I know for in cartesian coordinates: $$ \mathbf{E}(x,y,z)=P\mathbf{\hat{x}}+Q\...
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2answers
50 views

Natural logarithm question 1

i tried to derive logistic population model, and need to integrate this $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t$. here is my solution $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=\int \frac{...
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1answer
30 views

Evaluate $\int x^2dx$ using darboux sum

Let take the partition $P=\{0,\frac{1}{n},\frac{2}{n},...,\frac{n-1}{n},1\}$. Because $f(x)=x^2$ is strictly increasing $m_i$ we will be on the leftmost side of each interval and $M_i$ on the ...
3
votes
3answers
133 views

Help on how to show that $\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2$

$$\int_{0}^{1}\left(2{x-1\over \ln^2{x}}-{x+1\over \ln{x}}\right)dx=3\ln{2}-2\tag1$$ Rewrite, so we can apply Frullani's formula on first part $$\int_{0}^{1}\left(-{x+1\over \ln{x}}+{2\over \ln{x}}+{...
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votes
3answers
90 views

Why the Jacobian isnt always 1? [on hold]

We have $A=\iint {\rm dx}' {\rm d}y'=\iint G \,{\rm d}x\,{\rm d}y$, where the integral is over a region with area $A$ in the $xy$-plane and $G$ the Jacobian of the coordinate transformation $x\to x'$ ...
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0answers
31 views

Find the primitive functions in given intervals [on hold]

I can easy compute the indefinite integral, but I have a problem with understanding the condition and I don't know what to do next - I don't know how to find a primitive function in a given interval. ...
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0answers
13 views

Why in the definition of multiple integrals on subset $A\subset \mathbb{R}^n$ it is required that $A$ is measurable?

I'm new with the study of multiple integrals. I think I understood the topics of Peano–Jordan measure. A multiple integral is defined on a measurable (and limited) subset $A\subset \mathbb{R}^n$, ...
0
votes
2answers
28 views

Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
3
votes
1answer
82 views

How to solve $ \int\sqrt{e^x}$ - different approaches seem to yield different results

Approaching the following integral in different ways appears to yield different results: $\int \sqrt{e^x} dx$ Simplifying $\sqrt{e^x}$ to $(e^x)^\frac{1}{2}$ to $e^\frac{x}{2}$. Now, integrating ...
0
votes
1answer
21 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
1
vote
0answers
33 views

Is there a general method to go about deriving a definite integral for a given result?

I was reading a blog post earlier about the Sophomore's Dream and a question came to mind: Say we wanted to find a definite integral that gives the following result $$\sum_{n=1}^\infty \left(\frac{a}...
1
vote
0answers
44 views

Inverting Laplace Transform with Geometric Series: $\mathcal{L}C(t) = \frac{\mu e^{-\beta \tau}}{1-\mu e^{-\beta \tau}}$

Question Am I correctly resuming the series to invert this Laplace Transform? Specific points of interest are bullet pointed. The Laplace Transform of a function, $C(t)$, is given by, $$ F(\beta) = \...
0
votes
2answers
61 views

Integration involving roots, and trigonometry

Question $$\int_{0}^{2} (1-\cos t) \sqrt{2(1-\cos t)} \, dt$$ Trying to find a solution to this integral problem, substitution method doesn't seem to work, while using the GDC gives a long decimal ...
0
votes
2answers
52 views

how i can convert a summation index n to integral?

$$\sum\limits_{n = 0}^N {f(x,n).\delta (x - {(n-a)^{2/3}})} $$ with $\delta$ is Dirac delta function (the solution can be an approximate calculation). Please help me. Thanks!