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Today I saw a book in the bookstore that has the following integral on its cover:

$$\int \frac {dx}{dx} = \frac {1}{d} \ln x + c$$

I don't understand the meaning of $\frac {1}{d}$. Also, $\frac {dx}{dx}$ looks odd to me. Can someone please explain these notations to me? I think these notations may be valid, because the author is very famous in my country.

If this is a valid integral, can someone also tell me how to solve integrations that has more than one differentials? By "differential", I mean $dx$, $dy$, $du$, etc. For example:

$$\int \frac {u \cdot dx^2}{du^2} \tag 1 $$

P.S. The above integral is actually:

$$\int \frac {\sin(x)}{\cos^2(x)}dx \tag 2 $$

If you try to solve (2) by a variable change of $u = \sin(x)$, the outcome would be (1) (I know that (1) can be easily solved by getting $\cos^2(x)$ as $u$). But I want to know that if (2) has also an answer.

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    $\begingroup$ I think, $\int \frac{dx}{dx}$ is $\int \frac{dx}{d\cdot x}$. It's not a notation, it's bad using of $d$ $\endgroup$ – Michael Galuza Jul 18 '15 at 17:25
  • $\begingroup$ @MichaelGaluza Why not turn that into an answer? $\endgroup$ – Cameron Williams Jul 18 '15 at 17:27
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    $\begingroup$ What is the book? $\endgroup$ – Masacroso Jul 18 '15 at 18:04
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    $\begingroup$ It's probably a joke of the author on the cover. $\endgroup$ – Rogelio Molina Jul 18 '15 at 18:08
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    $\begingroup$ Thanks for all comments. @Masacroso, the book is "methods of integration" by "Ali Bayat Movahhed", however, the book is in Persian. The author's website is available at this link. The cover itself is available in low resolution at this link $\endgroup$ – ali akbar gholami Jul 18 '15 at 18:11
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The book cover is a mathematical pun. It's treating the $d$ in the denominator as a constant coefficient of $x$. So just as $$ \int \frac{dx}{A \cdot x} = \frac{1}{A} \ln |x| + C $$ you could write $$ \int \frac{dx}{d \cdot x} = \frac{1}{d} \ln |x| + C $$ except that there's no reason to do that unless you're being deliberately confusing.

In general, "integrals that have more than one differential" aren't meaningful in the way you want them to be.

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