# 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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### Is it possible to $\int \sqrt{\cot x}$ by hand

$$\int \sqrt{\cot x}{dx}$$ $$\int \sqrt{\frac{\cos x}{\sin x}}{dx}$$ Using half angle formula $$\int \sqrt{\frac{1-\tan^2 \frac{x}{2}}{2\tan \frac{x}{2}}}{dx}$$ But I am not getting any lead from ...
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### Integral Equation without solution?

working on a physical problem I arrived at the following equation $$y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
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### When an inner product of two continuous functions and one of them are given, how can I find the other one?

$f(p)$ is the inner product of $h(t,p)$ and $x(t)$. That is, $\int_{T} h(t,p)x(t)dt = f(p)$ When $h(t,p)$ and $f(p)$ are given, how can I find $x(t)$?
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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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### How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$\int\frac{dx}{(2\sin x+\sec x)^4} \ ?$$
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### Solution to the convoluted integral equation

A have the following equation: $$f(a)=\int_0^ag(x)f(x)\,dx,$$ where $g(x)$ is a known function. Is there any solution to $f(a)$ just in terms of $g$?
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### If an integrable function is orthogonal to all derivatives, then is f a constant?

Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that $$\int_{\mathbb{R}}f(x)v'(x)\,dx = 0$$ for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is ...
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### How to calculate limits in triple integral?

I have the next excersise of triple integrals: $f(x,y,z) = \cos(x + y + z)$, Limited by planes $x=\pi, y=\pi, z=\pi$ I have to find/determine the value for this. Specific Questions: How can I ...
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### 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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### 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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### Evaluate $\int \frac{\sqrt{64x^2-256}}{x}\,dx$

$$\int \frac{\sqrt{64x^2-256}}{x}\,dx$$ Image. I've tried this problem multiple times and cant seem to find where I made a mistake. If someone could please help explain where I went wrong I would ...
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### Why do I get two answer when calculating this integral from two ways?

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 ...
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### help me to find the triple integral

Use cylindrical coordinates to calculate for the given function and region: I found that the limits are for $x$ $0$ to $2\pi$ $r$ $0$ to $5$ and $z$ from $r^2$ to $25$ and the integration ...
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### 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 $f(x)$ ...
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### Is there a nice way to find this integral $\int_0^1\frac{ \arcsin x}{x} \mathrm{d}x$?

$$\int_0^1\frac{ \arcsin x}{x}\,\mathrm dx$$ I was looking in my calculus text by chance when I saw this example , the solution is written also but it uses very tricky methods for me ! I wonder If ...
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### Determine Integrability without use of Riemann Integral

Determine the function is integrable or not on its interval of definition: $f(x)=\begin{cases}0 \quad \textrm{if} \quad 0\le x\le1\\ x \quad \textrm{if} \quad 1\lt x \le2\end{cases}$ So in our class ...
$$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$$ $$\tan x + \cot x + \csc x + \sec x=\frac{\sin x + 1}{\cos x} +\frac{\cos x + 1}{\sin x}$$ $$= \frac{\sin x +\cos x +1}{\sin x \cos x}$$ $$t= \... 3answers 3k views ### How to solve an definite integral of floor valute function? How do you prove this identity:$$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$I'd very much appreciate your help on this one! 1answer 258 views +50 ### A tricky integral - \int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt$$ \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 $$where n ... 1answer 49 views ### Iterated Integral with variable substitution I need to calculate the double integral of the function f(x,y) = (x+y)^9(x-y)^9: \int_0^{1/2} \int_x^{1-x} (x+y)^9(x-y)^9 dydx I have a solution but I definitely arrived at it after a sloppy ... 1answer 144 views ### What is relation between these integrals I know$$ \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$$Please guid me. I have sixteen ... 1answer 67 views ### Volume of the ellipsoid (x+2y)^2+(x-2y+z)^2+3z^2=1 Find the volume of the ellipsoid (x+2y)^2+(x-2y+z)^2+3z^2=1, using integration. It is clear that this is not centered at the origin. So, how do I find the limits for an integral? Any suggestion ... 3answers 128 views ### Integral that makes square root of \frac{\pi}{2} [duplicate] My question is regarding a integral that´s giving me a huge headache. I want to show$$\int_{0}^{\infty}y^2e^{-\frac{y^2}{2}}dy=\sqrt{\frac{\pi}{2}}$$I'm studying for an exam. I'm suppose to find ... 3answers 108 views ### Nice way to solve \int\int \frac{1}{1-(xy)^2} dydx? This is something I've been thinking about lately;$$\int_0^1 \int_0^1 \frac{1}{1-(xy)^2} dydx$$Solutions I've read involve making the substitutions: x= \frac{sin(u)}{cos(v)} and y= \frac{sin(v)}... 2answers 60 views ### How to evaluate this integral \int_{-\infty}^{\infty}\exp(-ay^2)dy [duplicate] I want to evaluate this integral$$\int_{-\infty}^{\infty}\exp(-ay^2)dy$$using the error function definition. The problem I am facing is with the coefficient of y^2. Any suggestions? Fact$$...
$$\int{\sqrt{1-{x^3}}}dx$$ I tried with $t=x^3$ but then I have the $3x^2$ dt that I can't get rid of.