Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.93
Evaluate the integrals in Exercises 91–102.
93. ∫(arcsin x)²dx/√(1-x²)
Verified step by step guidance1
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.
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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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06:35Derivatives 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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07:32Representing Functions as Power Series
Related Practice
Textbook Question
43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.
Textbook Question
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