91Ó°ÊÓ

Trigonometric identities are mathematical equations that relate the angles and sides of a triangle with trigonometric functions. These identities help simplify complex trigonometric expressions and are fundamental tools in calculus. One useful identity is the double angle formula for cosine:

• \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \)

This identity makes it easier to manage integrals involving trigonometric functions. Given our original expression \( \frac{1}{1+\cos x} \), it can be rewritten using this identity, simplifying the integration process. Instead of wrestling with complex terms, you employ this rule to transform the integrand into a more manageable form. Using the identity to write it as \( \frac{1}{2}\sec^2\left(\frac{x}{2}\right) \) allows us to substitute more easily later on in the problem. Recognizing and applying these formulas effectively is a significant step in solving calculus problems involving trigonometry.

Definite integrals are employed to calculate the accumulated sum of areas under a curve, from one point to another on the x-axis. They are not simply determining the value of an anti-derivative but finding the net area between two defined points. In our problem, we had to compute:

• \( \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{1}{1+\cos x} \, \mathrm{d}x \)

The limits \( \frac{\pi}{4} \) and \( \frac{3\pi}{4} \) define the interval on the x-axis over which we calculate the area. Transforming the function into a simpler form using trigonometric identities and substitution allows us to handle the integral and evaluate it with these set boundaries.With definite integrals, after evaluating the antiderivative, the last step is always to apply the limits of integration to get the actual number, which represents the cumulative sum over the range given.

The substitution method, or "u-substitution," is a powerful technique in calculus that simplifies the process of integration. It involves changing variables to transform a complicated integral into a simpler, more manageable form. In our example, we used the substitution:

• Let \( u = \frac{x}{2} \)
• Thus, \( \mathrm{d}u = \frac{1}{2}\mathrm{d}x \) or \( \mathrm{d}x = 2 \, \mathrm{d}u \)

This substitution adjusts the integrand, making it \( \int \sec^2(u) \, \mathrm{d}u \), which is simpler to integrate.After substituting, it's necessary to also change the limits of integration according to the variable \( u \). When \( x = \frac{\pi}{4} \), \( u = \frac{\pi}{8} \), and when \( x = \frac{3\pi}{4} \), \( u = \frac{3\pi}{8} \). By redefining the bounds, we maintain the integrity of the definite integral.At this stage, integrating \( \sec^2(u) \) gives \( \tan(u) \), which is straightforward. Calculate this over the new limits to find the solution. The substitution method elegantly reduces workload, transforming integrals into expressions that are much easier to handle.