Integration of [tanx] dx Limit 0 to 5pi/2 ….. .p.s This is a gif function.............

To solve the integral of the tangent function, specifically \(\int_0^{\frac{5\pi}{2}} \tan(x) \, dx\), we need to consider the properties of the tangent function and its behavior over the specified interval. The tangent function has a periodicity of \(\pi\), which means it repeats every \(\pi\) units. This periodicity will play a crucial role in evaluating the integral.

Understanding the Tangent Function

The tangent function, \(\tan(x)\), can be expressed as the ratio of sine and cosine: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). It has vertical asymptotes where the cosine function equals zero, specifically at odd multiples of \(\frac{\pi}{2}\). Within the interval from \(0\) to \(\frac{5\pi}{2}\), we encounter these asymptotes at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\).

Breaking Down the Integral

Given the asymptotes, we need to split the integral into manageable parts. The integral can be expressed as:

• \(\int_0^{\frac{\pi}{2}} \tan(x) \, dx\)
• \(\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \tan(x) \, dx\)
• \(\int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} \tan(x) \, dx\)

Calculating Each Integral

Let's evaluate each part separately:

1. From 0 to \(\frac{\pi}{2}\)

The integral \(\int_0^{\frac{\pi}{2}} \tan(x) \, dx\) can be computed using the antiderivative of \(\tan(x)\), which is \(-\ln|\cos(x)|\). Thus:

\[ \int_0^{\frac{\pi}{2}} \tan(x) \, dx = \left[-\ln|\cos(x)|\right]_0^{\frac{\pi}{2}} = -\ln(0) - (-\ln(1)) = \infty \]

This integral diverges to infinity.

2. From \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\)

In this interval, \(\tan(x)\) also diverges because it approaches negative infinity at \(x = \frac{3\pi}{2}\). Therefore:

\[ \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \tan(x) \, dx = \infty \]

3. From \(\frac{3\pi}{2}\) to \(\frac{5\pi}{2}\)

Similarly, in this interval, \(\tan(x)\) diverges again as it approaches positive infinity at \(x = \frac{5\pi}{2}\):

\[ \int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} \tan(x) \, dx = \infty \]

Final Evaluation

Since each of the integrals diverges, we conclude that the overall integral \(\int_0^{\frac{5\pi}{2}} \tan(x) \, dx\) does not converge to a finite value. Instead, it diverges to infinity due to the behavior of the tangent function at the asymptotes within the interval.

In summary, the integral of \(\tan(x)\) from \(0\) to \(\frac{5\pi}{2}\) diverges, and thus we cannot assign it a finite value. This is a critical aspect to remember when dealing with integrals of functions that have vertical asymptotes within the limits of integration.