# Tagged Questions

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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### Final step in integration

I was solving a integration problem in which we have to integrate the following $$\int\frac{dx}{x\sqrt{x^2+1}}$$ I tried it lot , i think almost got the last step but the answer did not match . My ...
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### How to compute integral $\int_0^1\sqrt{1-x^2}dx$ without using trigonometric functions and also answer should not contain trigonometric functions

I am working on definite integrals, and not getting solution for above question, anyone can help me in this?
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### How do I justify this differentiating under the integral? (Complex “Wirtinger derivative”)

I would like to differentiate under the integral in this situation: $$\frac{d}{d\bar{z}}\int_0^\infty h(t)e^{tz}\,dt=\int_0^\infty \frac{d}{d\bar{z}}h(t)e^{tz}\,dt$$ where $\frac{d}{d\bar{z}}$ is ...
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### Does $\int^{\infty}_0 \frac{\cos x}{1+x}\,\mathrm dx$ converge?

The integral $$\int^{\infty}_0 \frac{\cos x}{1+x}\,\mathrm dx$$ is shown to be equal to $$\int^{\infty}_0 \frac{\sin x}{{(1+x)}^2}\,\mathrm dx$$ through integration by parts. The latter one converges ...
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### Integral substitution?

In a recent economics paper (Rhodes and Wilson 2016), expected buyer surplus is computed as follows (page 8, (4)): $$v^*(q) = \int_{p^*(q)}^{b+q}[1-G(z-q)] dz$$ where $G(\cdot)$ is a distribution ...
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### Finding the boundaries on a triple integral

Solve: $$\iiint yz \,dV$$ Over the tetrahedron with vertices on the points $$A(0,0,0), B(1,1,0), C(1,0,0), D(1,0,1)$$ Well, I proceeded to find a the equation of a plane which contained B, C and D. ...
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### F(x) = 0 for all points except c. Show F is integrable.

Suppose $c$ is a point in the closed $[a,b]$ and that $F(x) = 0$ for all $x$ in $[a,b]$ except for $c$ and that $F(c) = 1$. Show that $F$ is integrable on $[a,b]$ and that $\int_a^bF(x)dx = 0$. By ...
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### Elaboration on Lagrange's algebraic approach to calculus?

In lectures 4 and 8 of NJ Wildberger's differential geometry series, he presents an algebraic method of taking the derivative of a polynomial that was apparently devised by Lagrange. He briefly ...
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### Conditional mutual information for continuous random variables

Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more ...
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### Evaluation of $\int^{\pi/2}_{0} \frac{x \tan(x)}{\sec(x)+\tan(x)}dx$

Evaluate the given integral: $$\int^{\pi/2}_0 \frac{x \tan(x)}{\sec(x)+\tan(x)}dx$$ I multiplied and divided by $\sec(x)+\tan(x)$ to get denominator as $1$ but In calculation of integral, $x$ is ...
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### Help needed in solving integration

I want to solve the following integration $$\int_0^{\infty}[1-(\frac{1}{1+x})^M]x^{-\frac{2}{\alpha}-1}dx$$ where $M$ is a positive integer and $\alpha \geq 2$ My attempt: In my attempt I use the ...
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### $\int f(x)\,dx - \int f(x)\,dx$

which is true $$\int f(x)\,dx - \int f(x)\,dx = 0$$ or $$\int f(x)\,dx - \int f(x)\,dx=c\text{ ?}$$ with $c$ some arbitary constant. My intuition says that 'something' subtracted by itself is ...
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### Obatin $\int_{\gamma_1}F\cdot dl =\int_{\gamma_2}F\cdot dl$

Let $F = (F_1,F_2)$ be a $C^1$ vector field such that all its components are continuously differentiable in $\Omega$. Assume that $\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x}$ Let ...
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\begin{align} \frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\ \tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)} \end{align} So far so good. Now introduce a complex variable $z$ ...
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Suppose $A\subset \mathbb{R}^n$ is a compact, convex and centrally symmetric set such that $(x_1,\ldots,x_n)\in A$ if $$|x_1|+\ldots+|x_r|+2\left(\sqrt{x_{r+1}^2 + x_{r+2}^2} + \ldots + \sqrt{x_{n-1}^... 1answer 31 views ### Convergence of a integral for every curve in the sphere Let S be the unit open sphere in \mathbb{R}^3: x^2+y^2+z^2< 1 and \partial S its border x^2+y^2+z^2= 1. Let f:S\cup \partial S\rightarrow \mathbb{R} be a continuous function which is ... 2answers 49 views ### Evaluate \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} How do you solve this integral$$ \phi = \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} $$? Note: It appears in the Kepler problem and it should ... 1answer 23 views ### Choice of the limits for multivariable integral Let A \subseteq \mathbb{R}^2 a limited set bordered through x=0, x=1, y=-1+x, y=1-x^2. Rotate A around the y-axis and define this set with B. Calculate the integral$$\int_B y\,\mathrm{d}x\...
Assuming $a(t)=a_0\sin(\omega t)$, $v(0)=0$ and $x(0)=0$. I hope you know about basic relation between position, velocity and acceleration. They are derivatives of the proceeding one. I went on ...
To prove $\int x^p \, dx = \frac{x^{p+1}}{p+1} + C$, my calculus textbook writes: F '(x) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1} +C\right) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1}\right)+\frac{d}{...