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

Accepted Answer

Recognize that the integral involves trigonometric functions in both numerator and denominator, which suggests using the substitution $$u = 	an\left(\frac{	heta}{2}ight) to $$convert the integral into a rational function of $$u. $$Use the given substitution formulas:

$$dx = \frac{2}{1 + u$$^{2}$$} du$$,

$$\sin 	heta = \frac{2u}{1 + u$$^{2}$$}$$,

$$\cos 	heta = \frac{1 - u$$^{2}$$}{1 + u$$^{2}$$}.

$$Replace $$d	heta$$, $$\sin 	heta$$, and $$\cos 	heta in $$the integral accordingly. Rewrite the integral limits in terms of $$u$$:

When $$	heta = 0$$, $$u = 	an(0) = 0$$;

When $$	heta = \frac{\pi}{2}$$, $$u = 	an\left(\frac{\pi}{4}ight) = 1.

So $$the integral limits change from $$	heta \in [0, \frac{\pi}{2}] to$$ $$u \in [0, 1]. $$Substitute all parts into the integral:

$$\int_{0}^{\frac{\pi}{2}} \frac{d	heta}{\cos 	heta + \sin 	heta} = \int_{0}^{1} \frac{\frac{2}{1 + u$$^{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$$^{2}$$). $$Simplify the entire integrand to a rational function in $$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.

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주요 개념

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

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

Definite Integral Limits Transformation