Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.6.93

Chapter 7, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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