Integration by parts

We know from the product rule that $d(xy) = xdy+ydx$. Integrating it,
$$\int d(xy) = \int xdy + \int ydx$$
$$\therefore \int xdy = xy- \int ydx$$

This has applications in many different problems, and is known as integration by parts.

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The $\operatorname{erf}$ function

Can a function be 'defined' as the anti-derivative of another function? $\operatorname{erf}$ is one such function. Even though it has various proper infinite series expansions (as in Wolfram MathWorld - Erf ), it is defined by mathematicians as such: $$\operatorname{erf}(z) \equiv \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$ This function has extensive implications in statistics, and can be used to express the integral of $e^{-x^2}$ and $e^{x^2}$. See the post that tries to find integral of $e^{x^2}$ - we need not use integration by parts there, but rather can use the $\operatorname{erf}$ function. Source: Wolfram MathWorld

Why $y^2 = x^3 + 7$ has no integral solutions?

Consider the given equation... $$y^2-7=x^3=(x^{\frac{3}{2}})^2$$ $$y^2-(x^{\frac{3}{2}})^2=7$$ If $x^{\frac{3}{2}}$ is a decimal, then $y$ should also be some decimal to make the equation an integer, i.e., $7$. So $y$ will be a decimal in this case. But consider when $x^{\frac{3}{2}}$ is an integer. The only integers whose difference of squares is $7$, is $4$ and $3$ (why?). $$4^2 - 3^2 =7$$ But compare it to the equation $y^2-(x^{\frac{3}{2}})^2=7$. Then $$x^{\frac{3}{2}} = 3$$ $$x = 3^{\frac{2}{3}}$$ But here $x$ is a decimal! Therefore, at least one of the variables $x$ and $y$ is a decimal in the given equation, thus proving the fact that it has no integral solutions.

Is this seat taken?

100 people are waiting to board a plane. The first person’s ticket says Seat 1; the second person in line has a ticket that says Seat 2, and so on until the 100th person, whose ticket says Seat 100. The first person ignores the fact that his ticket says Seat 1, and randomly chooses one of the hundred seats (note: he might randomly choose to sit in Seat 1). From this point on, the next 98 people will always sit in their assigned seats if possible; if their seat is taken, they will randomly choose one of the remaining seats (after the first person, the second person takes a seat; after the second person, the third person takes a seat, and so on). What is the probability the 100th person sits in Seat 100? This problem can be solved intuitively. The first step to solve this problem is to understand that the last person will either get his seat or the first person's seat. But why? If the 1st person chooses 1st seat itself, everybody gets their own seat (i.e., the last person will g...

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