Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)

Accepted Answer

First, examine the integral: $$\int_{\pi/2}^{2\pi/3} \frac{\cos 	heta \, d	heta}{\sin 	heta \cos 	heta + \sin 	heta}. $$Notice the denominator can be factored to simplify the expression. Factor the denominator: $$\sin 	heta \cos 	heta + \sin 	heta = \sin 	heta (\cos 	heta + 1). So $$the integral becomes $$\int_{\pi/2}^{2\pi/3} \frac{\cos 	heta}{\sin 	heta (\cos 	heta + 1)} \, d	heta. $$Consider a substitution to simplify the integral. Since both $$\sin 	heta$$ and $$\cos 	heta$$ appear, try substituting $$u = \sin 	heta or$$ $$u = \cos 	heta + 1. $$Calculate $$du$$ accordingly to see which substitution simplifies the integral best. If you choose $$u = \cos 	heta + 1$$, then $$du = -\sin 	heta \, d	heta. $$Rearranging, $$-du = \sin 	heta \, d	heta. $$This matches part of the denominator and the differential, so rewrite the integral in terms of $$u. $$Rewrite the integral using the substitution and adjust the limits of integration accordingly by plugging in the original $$	heta$$ limits into $$u = \cos 	heta + 1. $$Then, express the integral fully in terms of $$u$$ and integrate.

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Key Concepts

Trigonometric Substitution

Definite Integrals and Limits of Integration

Simplification of Rational Trigonometric Expressions