# Tagged Questions

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### Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
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### The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
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### Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
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### fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
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### Is the definite integral of the function necessarily the anti-derivative?

Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$ By the integral definition, g(x) is the area under the curve of f(x) from 1 to x. eg: g(5) is the area under f(x) from 1 to 5. I ...
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### How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
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### I need to find know how to integrate $x$ multiplied by a function to a power that is a fraction.

I know how to find integral functions normally, but when I try to find it from say $x\sqrt{4-x^2}$, I get completely lost.This screws me up in both indefinite and definite integration, so please help
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### Why should we get rid of indefinite integration?

It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that. ...
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### Integral of $\frac{1}{x\ln(x+1)}$

I'm trying to get my head around calculating $$\int\frac{1}{x\ln(x+1)}dx.$$ I can't seem to get anywhere. I tried parts and substitutions, but that $(x+1)$ is always in the way. Any suggestions? ...
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### Uses of integral calculus in discrete mathematics?

I have to do a project in my integral calculus class. But all the topics are too mainstream (parabolic arc calculation,archimedean approzimation of circle are,obtaining $E=mc^2\dots$ However I'm ...
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### Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
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### Are all antiderivatives of a function an area or a (difference of areas) function of its derivative?

If you take $f(t)=2t$ then the integral $\int_a^xf(t)dt=x^2-a^2$ seems to be the general antiderative of $f$ and the notation of $\int f(t)dt$ makes sense as it is saying the antideritave of $f$ is ...
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### Explain me definite integral…

So, as the derivative is a tangent of an angle between the $x$-axis and the corresponding tangent line, how can we represent an indefinite integral, and why is the area (definite integral) of, for ...
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### Find recursive forumula for integrals

I have to find recursve formulas for solving the following two integrals. The assignment tells one to find an Expression that leads from the calculation of $\dfrac{I_{2n}}{I_{2n+1}}$ to the ...
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### Definite integration problem (trig).

I have this definite integral: $$\int_0^\Pi \cos{x} \sqrt{\cos{x}+1} \, dx$$ For finding the indefinite integral, I have tried substitution, integration by parts, but I'm having trouble solving ...
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### an intriguing integral $I=\int\limits_{0}^{4} \frac{dx}{4+2^x}$

I solved this integral: $$I=\int\limits_{0}^{4} \frac{dx}{4+2^x}$$ In a method similar to the one used in An interesting integral $I = \int\limits_{-1}^{1} \arctan(e^x)dx$. This integral ...
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### Express the indefinite integral $\int\sin x^2~dx$ as a power series

What does this mean? I never saw this in my class/notes so I don't understand the conversion from integral to power series. Also if the integral were defined from $0$ to $1$, what new steps do I add? ...
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### Indefinite integral with multiple values evaluated at endpoints to get definite integral

I am having trouble evaluating the definite integral $$\int_{-\pi}^{\pi} \frac{\sin\phi}{\pi\left( 1-\cos^2\Lambda\cos^2\phi \right)}d\Lambda$$ by the standard technique of the difference of value of ...
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### Find point between two intervals which equals the average value.

Find the point in the interval [5,9] at which the function $f(x)=17e^{3x}$ equals its average value on that interval. So I've got my function as follows ...
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