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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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Carrying out this integral

I am struggling to reproduce the following result. Given- $$I_1 = \int_{y=-\infty}^\infty \int_{y'=-\infty}^\infty\int_{x=-\infty}^\infty \int_{x'=-\infty}^\infty dy dy' dx dx' \frac{n(x) n(x')}{[(x-x')^2 + (y - y')^2]^{3/2}}$$ and $$I_2 =…
rahman62
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Need hints for the following integrals

I am working on finding a surface integral solution for caculating an arbitrary point magnet field $\vec{B}$ distributed by a disc magnet via magnetic surface charge model. I used the assumption that surface charge is uniform. After I wrote out and…
Elder Yang
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Evaluating an integral with a logarithmic singularity

The function I'm looking to find is: $$ \sigma = \int_{-a}^{a} \frac{(x-x')((x-x')^2 - y^2)}{((x-x')^2 + y^2)^2} \ln \left|\frac{x(a^2 - c^2)^{1/2} + c(a^2 - x^2)^{1/2}}{x(a^2 - c^2)^{1/2} - c(a^2 - x^2)^{1/2}}\right| dx$$ Analytically there aren't…
tramninety
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How to calculate this complex definite integral

$$\int_0^{t_0}\frac{518.4935}{{66.9}-\frac{3.4A^2e^{at-at_0}}{0.029Ae^{at_0}+0.008A^3e^{3at_0}-0.726}}\mathrm{d}t$$ I calculated it into the following formula, but I don't know whether it is right or not, and I don't know how to calculate…
用户1084369
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Best Method for computing the definite Integral of this trigonometric function

$$ \int_0^{10^{50}}\sin(10^{20} \sqrt{x^2 + 10^{100}})dx$$ The interval is large and I have problem seeing how to calculate it with the common numerical integral methods as they require taking lot of points across the function. This differs from the…
Smithy
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Order of definite integral bounds

Does the expression: $$ \int_a^b f(x) dx $$ make any sense when a > b? Consider the simple case: $$ \int_1^0 x^2 dx = - 1/3 $$ But this of course makes no sense since area under a positive function like x^2 can not be negative! But if it doesn't…
Giack_89
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How to find constant for feynman's technique of integration $\int_{0}^{\infty}\frac{\ln\left(x^{2}+1\right)}{x^{2}+1}dx$

I've got an integral $$\int_{0}^{\infty}\frac{\ln\left(x^{2}+1\right)}{x^{2}+1}dx$$ and when I used Feynman's technique of integration $$I(t) = \int_{0}^{\infty}\frac{\ln\left(x^{2}+t\right)}{x^{2}+1}dx$$ I got the…
Bruh
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Require help evaluating a complicated definite integral

In my work I came across the following integral which stems from basically computing the statistics of an output signal of a nonlinear system given white noise at the input, so the integral is (if I did not make a mistake) represent a…
StrictlyStationaryPoster
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Relating two definite integrals

Let $$\int_0^{\infty}\frac{dx}{1+x^{2021}}=\int_0^1\frac{dx}{(1-x^{2021})^{a}}$$ then find $a$ This is one of the questions of the HOTS section of my practice sheet. I can not think of any way to solve it. I got some hint from limits of the…
Acc2
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Taking an Integral with respect to time to find the work done on a rocket.

I was given a rocket problem for my Calculus 2 class and want to know if it's possible to take an integral of time to find the correct solution. I think it will be easier to explain if I show the problem and expected solution first. Here is the…
StudentOfMorpheus
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Evaluating $\int_0^a \int_0^{a-x} e^{y(2a-y)} dy dx $

I = $\int_0^a \int_0^{a-x} e^{y(2a-y)} dy dx $ I have questions in calculating this one, and the reference answer does this: I = $\int_0^a \int_0^{a-y} e^{y(2a-y)} dx dy $ questions solved: This applies the very basic trick to solve this double…
Veto Vote
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Does $\int_{-2}^{1}\frac{1}{x^2}dx=-\frac{3}{2}$?

My thinking started when I got the answer to below integral $$\int_{-2}^{1}\frac{1}{x^2}dx$$ I normally found the anti-derivative as $\frac{-1}{x}$ and just substituted the limit. The answer I was getting is $\frac{-3}{2}$. Suddenly I thought "why…
Rayon Cherry
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How can I compute this:$I=\int_0^\infty \frac{\ln(2x)^2 \cosh x} {1+\cosh 2x} dx$

How can I solve this integral? $$I=\int_0^\infty \frac{\ln(2x)^2 \cosh x} {1+\cosh 2x} dx$$ I get this: $I=I_1+I_2$ and $I_1=\int_0^\infty\frac{\ln 2}{e^x+e^{-x}}dx=\pi/2$ and $I_2=\int_0^\infty\frac{\ln x}{e^x+e^{-2x}}dx$ and I can't solve $I_2$.…
Costas
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Integral using cylindrical coordinates

Is it fine? Using cylindrical coordinates to compute $$\iiint_T (x^2+y+z^2)^3 dV $$ where $T$ is the solid $x^2+z^2=1$, $y=0$, $y=2$, $z=0$. Attempt: using cylindrical coordinates $$\int_0^{\pi/2}\int_0^2\int_0^{\sqrt{1-r^2\cos^2\theta}}…
apa
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How to evaluate: $I=\int_{0}^{2\pi}\frac{d\theta}{\sqrt{k-\cos\theta}}$

Let $k>1$. How to evaluate: $$I=\int_{0}^{2\pi}\frac{d\theta}{\sqrt{k-\cos\theta}}$$ Since $\cos\theta$ is even around $\theta=\pi$, we have: $$I=2\int_{0}^{\pi}\frac{d\theta}{\sqrt{k-\cos\theta}}$$ I use $x=\cos\theta$, then:…
Aria
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