Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ (1 - x²)^(1/2) / x⁴ dx

Accepted Answer

Identify the integral to solve: $$\int \frac{\sqrt{1 - x$$^{2}$$}}{x$$^{4}$$} \, dx. $$Recognize that the integrand contains $$\sqrt{1 - x$$^{2}$$}$$, which suggests a trigonometric substitution using $$x = \sin 	heta$$ because $$1 - \sin$$^{2}$$ 	heta = \cos$$^{2}$$ 	heta. $$Make the substitution $$x = \sin 	heta$$, then compute $$dx = \cos 	heta \, d	heta. $$Rewrite the integral in terms of $$	heta$$: $$\int \frac{\sqrt{1 - \sin^{2} 	heta}}{\sin^{4} 	heta} \cdot \cos 	heta \, d	heta = \int \frac{\cos 	heta}{\sin^{4} 	heta} \cdot \cos 	heta \, d	heta = \int \frac{\cos^{2} 	heta}{\sin^{4} 	heta} \, d	heta.$$ Simplify the integral to $$\int \frac{\cos$$^{2}$$ 	heta}{\sin$$^{4}$$ 	heta} \, d	heta. $$Use the identity $$\cos$$^{2}$$ 	heta = 1 - \sin$$^{2}$$ 	heta to $$rewrite the numerator if needed, and express the integral in terms of powers of $$\sin 	heta to $$facilitate integration. After integrating with respect to $$	heta$$, substitute back $$	heta = \arcsin x to $$express the answer in terms of $$x.$$

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - x²)^(1/2) / x⁴ dx

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Key Concepts

Trigonometric Substitution

Integration of Rational Functions

Simplifying Radicals in Integrals