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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.AAE.3
Capitolo 8, Problema 8.AAE.3

Evaluate the integrals in Exercises 1–6.
∫ x arcsin x dx

Guida verificata passo dopo passo
1
Identify the integral to solve: \(\int x \arcsin x \, dx\).
Use integration by parts, where you let \(u = \arcsin x\) and \(dv = x \, dx\).
Compute \(du\) by differentiating \(u\): \(du = \frac{1}{\sqrt{1 - x^2}} \, dx\), and compute \(v\) by integrating \(dv\): \(v = \frac{x^2}{2}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int x \arcsin x \, dx = \frac{x^2}{2} \arcsin x - \int \frac{x^2}{2} \cdot \frac{1}{\sqrt{1 - x^2}} \, dx\).
Simplify the remaining integral \(\int \frac{x^2}{2 \sqrt{1 - x^2}} \, dx\) and consider substitution methods to evaluate it.

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Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.
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Integration by Parts for Definite Integrals

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin(x), are the inverses of the standard trigonometric functions. Understanding their derivatives and integrals is essential, as arcsin(x) has a derivative of 1/√(1 - x²), which often appears in integration problems involving these functions.
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Derivatives of Other Inverse Trigonometric Functions

Algebraic Manipulation in Integration

Algebraic manipulation involves rewriting integrals into simpler or more recognizable forms. This can include substitution, simplifying expressions, or breaking integrals into parts, which helps in applying integration techniques effectively and solving complex integrals step-by-step.
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