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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Stuck in Integretion while deriving Maxwell's Distribution

I've got this expression $$\int_0^{\infty} 4 \pi v^2 C e^{-\frac{mv^{2}}{2kT}} \, dv$$ from 0 to ∞. I've tried everything from by parts to tabular but couldn't get anywhere. Is this even integrable?
Foon
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How to calculate this Stieltjes integral?

Given that f is a continuous function and [x] is the biggest integer which is smaller than x, I want to integrate f(x)d[x] from 0 to n, where n is a natural number. I know the definition of the Riemann-Stieltjes integral, but I can't actually use…
Math-Nerd
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What is the integral, $\int\frac{dx}{x + \sqrt{1-x²}}\ $?

What is the integral, $$\int\frac{dx}{x + \sqrt{1-x²}}\ ?$$
Aiden Sherry
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How to find the area bounded by the curves $x=a\cos t$ and $y=b\sin t$ in the first quadrant?

I know the given equations are the parametric equation of an ellipse . The curve meet the x axis at $(a,0)$ in the first quadrant . Now I do this $\int_{0}^{a} y dx$ My book has the following step which I am unable to…
Seeker1201
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What happens when the interval of an integral changes from infinity to a constant number?

There exist a calculation about electromagnetic mass: $$m_\mathrm{em} = \int {1\over 2}E^2 \, dV = \int\limits_{r_e}^\infty \frac{1}{2} \left( {q\over 4\pi r^2} \right)^2 4\pi r^2 \, dr = {q^2 \over 8\pi r_e}$$ Reducing $r_e$ we get infinite mass.…
HolgerFiedler
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Definite integral of exponential function $se^{s^{2}}$

I'm trying to figure out how to solve this but haven't come up with anything yet: $$\int\limits_0^t{se^{s^{2}}}ds$$ The solution I wrote down is: $$\frac{1}{2}(e^{t^{2}}-1)$$ Can anyone tell me how I can get to this solution? Thanks a lot.
ocram
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Evaluating a double integral that arises in computing the solid angle subtended by an equilateral triangle

I was evaluating an double integral with following limits. $$\Omega =\int_{-\frac{a}{2\sqrt{3}}}^{\frac{a}{\sqrt{3}}} \int_{-\frac{a-\sqrt{3}y}{3}}^{\frac{a-\sqrt{3}‌​y}{3}} \frac{a\,dy\,dx}{(x^2+y^2+a^2)^{3/2}}$$ (It's solid angle subtended by an…
Someone
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Why does the integral of absolute value function not return actual area?

Suppose my function is f(x)=$x^2-1$. The absolute value of this function is $\sqrt{(x^2-1)^2}$. So why doesn't the area of this function between -2 and 2 equal $\int_{-2}^2\sqrt{(x^2-1)^2}$? The antiderivative exists and equals…
C Shreve
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Finding a function

I have the value of a function's integral and two points of this function. I'm searching for the function. Is it possible to find it? $$\int F dx= a$$ $$F(b)= F(c)= d$$ $a$, $b$, $c$ and $d$ are known. I'm searching for $F$
krestin
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What is solution for the following integral function

The following integral is taken from exercise 7.1 of james stewart calculus 7th edition early transcendentals $$\int_0 ^te^s\sin(t-s)~ds$$ I tried integration by parts and got the answer as follows $-\frac{1}{2}[e^t+\cos(t)+\sin(t)]$. Please verify…
joe1983
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Given $s=\int_0^33t\sqrt{t^2+16}\,dt$ show that $s=61$

I decided to use U-substitution to show that $s=61$ but have failed. Here is my working, where have I gone wrong? Regards…
Thomas Winkworth
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Problem understanding Integration question

I do not understand the second task. The first task is evaluating following integral, what I did and got $-2π$ $$ \int_0^{2\pi}x\sin(x)\,dx=-2\pi\approx-6.2832. $$ However the second question is this: Find the area between the curve $y=x \sin x$ and…
user164612
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integration of definite integral

Evaluate $$\int\limits_{-1}^{\frac 3 2}|x\sin(\pi x)| dx $$ Can you help me find the domains in which the function will be positive and in which it be negative? I mean how to determine in which range it will positive or negative?
Maini who
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Splitting a definite integral doubt

Evaluate the integral $\int_0^{\pi}\frac{x}{a^2*\cos^2(x) + b^2*\sin^2(x)}\; dx$ In my textbook solution the integral has been split into intervals from $0$ to $\pi/4$ and then from $\pi/4$ to $\pi/2$. My question is what is the need to this? Why…
user34304
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Definite integral property

I need to prove or disprove this fact If $f$ is integrable on $[a,b]$, then there exists a number $c$ in the open interval $(a,b)$ such that $$\int_a^c f(x) \, dx = \int_c^b f(x) \,dx$$ I have tried to use this property $$\int_a^b f(x) \, dx =…
Usser123456798
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