87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.
A: dx = 2/(1 + u²) du
B: sin x = 2u/(1 + u²)
C: cos x = (1 - u²)/(1 + u²)
88. Evaluate ∫ dx/(2 + cos x).

Verified step by step guidance

1. Identify the substitution to simplify the integral: let \( u = \tan\left(\frac{x}{2}\right) \). This substitution transforms trigonometric expressions into rational functions of \( u \).
2. Express \( dx \), \( \sin x \), and \( \cos x \) in terms of \( u \) using the given formulas: \[ dx = \frac{2}{1 + u^{2}} \, du, \quad \sin x = \frac{2u}{1 + u^{2}}, \quad \cos x = \frac{1 - u^{2}}{1 + u^{2}} \]
3. Rewrite the integral \( \int \frac{dx}{2 + \cos x} \) by substituting \( \cos x \) and \( dx \) with their expressions in terms of \( u \): \[ \int \frac{dx}{2 + \cos x} = \int \frac{\frac{2}{1 + u^{2}} \, du}{2 + \frac{1 - u^{2}}{1 + u^{2}}} \]
4. Simplify the denominator by combining the terms over a common denominator \( 1 + u^{2} \): \[ 2 + \frac{1 - u^{2}}{1 + u^{2}} = \frac{2(1 + u^{2}) + (1 - u^{2})}{1 + u^{2}} = \frac{2 + 2u^{2} + 1 - u^{2}}{1 + u^{2}} = \frac{3 + u^{2}}{1 + u^{2}} \]
5. Substitute this back into the integral and simplify the fraction: \[ \int \frac{\frac{2}{1 + u^{2}} \, du}{\frac{3 + u^{2}}{1 + u^{2}}} = \int \frac{2}{1 + u^{2}} \times \frac{1 + u^{2}}{3 + u^{2}} \, du = \int \frac{2}{3 + u^{2}} \, du \] Now the integral is a rational function in \( u \), which can be integrated using standard techniques.

Key Concepts

Weierstrass Substitution (t = tan(x/2))

The Weierstrass substitution transforms trigonometric integrals into rational functions by setting u = tan(x/2). This substitution simplifies expressions involving sin x and cos x into rational functions of u, making integration more straightforward. It also changes the differential dx into a rational expression in terms of du.

Trigonometric Identities for sin x and cos x in terms of u

Using the substitution u = tan(x/2), sin x and cos x can be expressed as sin x = 2u/(1 + u²) and cos x = (1 - u²)/(1 + u²). These identities allow rewriting the integrand entirely in terms of u, facilitating the conversion of the integral into a rational function integral.

Integration of Rational Functions

After substitution, the integral becomes a rational function in u, which can be integrated using algebraic techniques such as partial fraction decomposition or standard integral formulas. Mastery of integrating rational functions is essential to solve the transformed integral efficiently.