To blow some freshman-minds away....

[mathbalarka]

Problem : Integrate $\displaystyle \int 1 \, \mathrm{d}x$.​

First, let $y = \displaystyle \int 1 \, \mathrm{d}x$. Then $\displaystyle \frac{dy}{dx} = 1$. By the geometric interpretation of differentiation, $\displaystyle \frac{dy}{dx}$ is $\tan(\theta)$, where $\theta$ is the angle made by the tangent at some $y = y(x)$ with the x-axis. In this case, $\tan(\theta) = 1$ is stationary, i.e., invariant under the change of $x$, so the function $y$ must coincide with the tangent made at any point on $y(x)$. Hence $y(x)$ is a linear function. Furthermore, $\tan(\theta) = 1$ is possible iff $\theta = \pi/4$, hence the tangent is the $45^\circ$-cut along the xy-plane, i.e., is the straightline $y' = x$. Then $y = y(x)$ must be parallel to this line, i.e., $y = x + C$ for some arbitrary real $C$. $\blacksquare$

 

[lofgran]

While the solution presented in the forum post is technically correct, it may not be the most intuitive or efficient way to approach this problem. it is important to consider multiple methods and perspectives when solving a problem.

One straightforward approach to integrating $\displaystyle \int 1 \, \mathrm{d}x$ is to use the basic rule of integration, which states that the integral of a constant is equal to that constant multiplied by the variable of integration. In this case, the constant is 1 and the variable of integration is $x$, so we have $\displaystyle \int 1 \, \mathrm{d}x = x + C$, where $C$ is the constant of integration.

Another approach is to think about the geometric interpretation of integration. The integral of a function represents the area under the curve of that function. In the case of $\displaystyle \int 1 \, \mathrm{d}x$, the function is simply a constant value of 1. This means that the area under the curve is just a rectangle with a base of $x$ and a height of 1. Therefore, the integral is simply the area of the rectangle, which is $x + C$.

In both of these approaches, we arrive at the same solution, but they may be more intuitive and easier to understand for someone who is not familiar with the geometric interpretation of differentiation or the concept of tangents. it is important to be able to communicate complex ideas in a clear and understandable way, and considering different approaches to a problem can help with this.