Question

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).

Accepted Answer

Identify the substitution to simplify the integral: let $$ u = 	an\left(\frac{x}{2}ight) . $$This substitution transforms trigonometric expressions into rational functions of $$ u . $$Express $$ dx $$, $$ \sin x $$, and $$ \cos x  in $$terms of $$ u $$ using the given formulas:

$$ dx = \frac{2}{1 + u^{2}} \, du, \quad \sin x = \frac{2u}{1 + u^{2}}, \quad \cos x = \frac{1 - u^{2}}{1 + u^{2}} $$ Rewrite the integral $$ \int \frac{dx}{2 + \cos x}  by $$substituting $$ \cos x $$ and $$ dx $$ with their expressions in terms of $$ u $$:

$$ \int \frac{dx}{2 + \cos x} = \int \frac{\frac{2}{1 + u^{2}} \, du}{2 + \frac{1 - u^{2}}{1 + u^{2}}} $$ Simplify the denominator by combining the terms over a common denominator $$ 1 + u^{2} $$:

$$ 2 + \frac{1 - u^{2}}{1 + u^{2}} = \frac{2(1 + u^{2}) + (1 - u^{2})}{1 + u^{2}} = \frac{2 + 2u^{2} + 1 - u^{2}}{1 + u^{2}} = \frac{3 + u^{2}}{1 + u^{2}} $$ Substitute this back into the integral and simplify the fraction:

$$ \int \frac{\frac{2}{1 + u^{2}} \, du}{\frac{3 + u^{2}}{1 + u^{2}}} = \int \frac{2}{1 + u^{2}} 	imes \frac{1 + u^{2}}{3 + u^{2}} \, du = \int \frac{2}{3 + u^{2}} \, du $$

Now the integral is a rational function in $$ u $$, which can be integrated using standard techniques.

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Concetti chiave

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

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

Integration of Rational Functions