Ch. 7 - Transcendental Functions

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Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.6.93

Capitolo 7, Problema 7.6.93

Evaluate the integrals in Exercises 91–102.
93. ∫(arcsin x)²dx/√(1-x²)

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Recognize that the integral is of the form \(\int \frac{(\arcsin x)^2}{\sqrt{1 - x^2}} \, dx\). Notice that the denominator \(\sqrt{1 - x^2}\) is the derivative of \(\arcsin x\) with respect to \(x\) because \(\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}\).

Use the substitution method by letting \(t = \arcsin x\). Then, differentiate both sides to find \(dt = \frac{1}{\sqrt{1 - x^2}} \, dx\), which implies \(dx = \sqrt{1 - x^2} \, dt\).

Rewrite the integral in terms of \(t\): since \(\arcsin x = t\), the numerator becomes \(t^2\), and the denominator times \(dx\) becomes \(\frac{1}{\sqrt{1 - x^2}} \times dx = dt\). Therefore, the integral simplifies to \(\int t^2 \, dt\).

Integrate \(\int t^2 \, dt\) using the power rule for integration: \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\). Here, \(n=2\), so the integral becomes \(\frac{t^3}{3} + C\).

Finally, substitute back \(t = \arcsin x\) to express the answer in terms of \(x\): the integral is \(\frac{(\arcsin x)^3}{3} + C\).

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Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. For the integral of (arcsin x)² / √(1 - x²), recognizing that the derivative of arcsin x is 1/√(1 - x²) suggests substituting t = arcsin x to transform the integral into a simpler form.

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Substitution With an Extra Variable

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin x, are the inverses of trigonometric functions and have specific derivatives and integrals. Understanding that d/dx (arcsin x) = 1/√(1 - x²) is crucial for manipulating integrals involving arcsin x.

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Derivatives of Other Inverse Trigonometric Functions

Integration of Powers of Functions

Integrating powers of functions, such as (arcsin x)², often requires techniques like substitution or integration by parts. Recognizing when to apply these methods helps in evaluating integrals involving squared inverse trigonometric functions.

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Representing Functions as Power Series

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