Final answer:
The integral of tan(x) diverges as x approaches π/2 because tan(x) becomes infinite at π/2, leading to an unbounded area under the curve. This divergent integral indicates an indefinite area, which is a result of tan(x)'s vertical asymptotes at odd multiples of π/2.
Step-by-step explanation:
Understanding the Divergence of an Integral
The question pertains to why the integral of tan(x) diverges as x approaches π/2. The tangent function, represented by tan(x), is known for its periodic behavior where it repeats every π radians (180 degrees). However, one of its most notable attributes are the points at which it becomes undefined, which are at odd multiples of π/2 (90 degrees).
As x approaches π/2 from either side, tan(x) heads towards positive or negative infinity, depending on the direction from which x approaches. This behavior is what causes the integral of tan(x) to diverge as it approaches π/2. In calculus terms, a divergent integral indicates that the area under the curve of the graph of the function is indefinite, and the function does not have a finite antiderivative.
To thoroughly comprehend this concept, consider what happens as we try to compute the integral from any number a to π/2 (where a < π/2). The integral calculates the area under the curve of tan(x) from x = a to x = π/2, and as we get closer to this upper limit, the value of tan(x) grows without bounds. Essentially, no matter how close to π/2 we choose to stop, we can always find a larger value of tan(x), implying that the area under the curve is unbounded.
In contrast, the integrals of even and odd functions over symmetric bounds can sometimes be straightforward to compute due to symmetry. For example, an odd function's integral over symmetry about the y-axis, such as from -L to L, will result in zero since the areas on both sides of the y-axis cancel each other out.
Ultimately, it is the behavior of tan(x) near its vertical asymptotes (odd multiples of π/2) that leads to the divergence of its integral when those points are within the limits of integration.