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)
91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).

Verified step by step guidance

1. Recognize that the integral involves trigonometric functions in both numerator and denominator, which suggests using the substitution \(u = \tan\left(\frac{\theta}{2}\right)\) to convert the integral into a rational function of \(u\).
2. Use the given substitution formulas: \(dx = \frac{2}{1 + u^{2}} du\), \(\sin \theta = \frac{2u}{1 + u^{2}}\), \(\cos \theta = \frac{1 - u^{2}}{1 + u^{2}}\). Replace \(d\theta\), \(\sin \theta\), and \(\cos \theta\) in the integral accordingly.
3. Rewrite the integral limits in terms of \(u\): When \(\theta = 0\), \(u = \tan(0) = 0\); When \(\theta = \frac{\pi}{2}\), \(u = \tan\left(\frac{\pi}{4}\right) = 1\). So the integral limits change from \(\theta \in [0, \frac{\pi}{2}]\) to \(u \in [0, 1]\).
4. Substitute all parts into the integral: \(\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\cos \theta + \sin \theta} = \int_{0}^{1} \frac{\frac{2}{1 + u^{2}} du}{\frac{1 - u^{2}}{1 + u^{2}} + \frac{2u}{1 + u^{2}}}\). Simplify the denominator by combining the fractions over the common denominator \((1 + u^{2})\).
5. Simplify the entire integrand to a rational function in \(u\), then integrate with respect to \(u\) over the interval \([0,1]\). Finally, evaluate the resulting expression at the limits to find the value of the integral.

Key Concepts

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

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

Trigonometric Identities for sin x and cos x in terms of tan(x/2)

Using the substitution t = tan(x/2), sine and cosine can be expressed as sin x = 2t/(1 + t²) and cos x = (1 - t²)/(1 + t²). These identities allow rewriting the integrand entirely in terms of t, facilitating the conversion of the integral into a rational function integral.

Definite Integral Limits Transformation

When substituting t = tan(x/2), the original integral limits in x must be converted to corresponding limits in t by evaluating t at the endpoints. For example, if x goes from 0 to π/2, then t changes from tan(0) = 0 to tan(π/4) = 1. Correctly transforming limits is essential for evaluating definite integrals after substitution.