How does one integrate the following: [tex] \int\limits^a_b

Ответ:

blakemccain1928

12.03.2022

Consider substituting u=4+xy

, so that $\mathrm du=x\,\mathrm dy$ . Then

$\displaystyle\int_{x=a}^{x=b}\int_{y=a}^{y=b}\frac x{(4+xy)^2}\,\mathrm dy\,\mathrm dx=\int_{x=a}^{x=b}\int_{u=4+ax}^{u=4+bx}\frac1{u^2}\,\mathrm du\,\mathrm dx$
$=\displaystyle\int_{x=a}^{x=b}\left(-\frac1u\right)\bigg|_{u=4+ax}^{u=4+bx}\,\mathrm dx$
$=\displaystyle\int_{x=a}^{x=b}\left(\frac1{4+ax}-\frac1{4+bx}\right)\,\mathrm dx$

Then by similar substitutions, you can easily find that you end up with

$\dfrac1a\ln|4+ax|-\dfrac1b\ln|4+bx|\bigg|_{x=a}^{x=b}$
$=\dfrac1a\ln|4+ab|-\dfrac1b\ln|4+b^2|-\dfrac1a\ln|4+a^2|+\dfrac1b\ln|4+ab|$
$=\dfrac1a\ln\left|\dfrac{4+ab}{4+a^2}\right|+\dfrac1b\ln\left|\dfrac{4+ab}{4+b^2}\right|$

Of course, this all assumes that the integrand is continuous over the domain of integration, which would require that a,b

are chosen such that $xy\neq-4$ for any $(x,y)\in[a,b]^2$ . If in particular ab-4

, then we can write

$=\dfrac1a\ln\dfrac{4+ab}{4+a^2}+\dfrac1b\ln\dfrac{4+ab}{4+b^2}$

and you can combine the logarithms if you like as

$=\ln\sqrt[a]{\dfrac{4+ab}{4+a^2}}+\ln\sqrt[b]{\dfrac{4+ab}{4+b^2}}$
$=\ln\left(\sqrt[a]{\dfrac{4+ab}{4+a^2}}\sqrt[b]{\dfrac{4+ab}{4+b^2}}\right)$

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Ответ:

Nevaeh3700

24.04.2022

Mathematics

Step-by-step explanation:

If you take the 100+850= 950 then you take that because that is the expenses and what she wants to make you will then divide 950/38=25 so sandra will need to sell 25 pie in order to cover the monthly cost and what she wants to earn.

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How does one integrate the following: $\int\limits^a_b \int\limits^a_b { \frac{x}{(4+xy)^{2} } } \, dy \, dx$ you can switch the bounds however you want (integrate with respect to x before y). i would like to know the best way to handle this problem.

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How does one integrate the following: you can switch the bounds however you want (integrate with respect to x before y). i would like to know the best way to handle this problem.

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How does one integrate the following: $\int\limits^a_b \int\limits^a_b { \frac{x}{(4+xy)^{2} } } \, dy \, dx$ you can switch the bounds however you want (integrate with respect to x before y). i would like to know the best way to handle this problem.