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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 8.2.82
Capitolo 8, Problema 8.2.82

Equations (4) and (5) lead to different formulas for the integral of arctan x:
a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]
b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]
Can both integrations be correct? Explain.

Guida verificata passo dopo passo
1
Step 1: Understand that both formulas represent the integral \( \int \arctan x \, dx \) but are expressed differently. Our goal is to verify if these two expressions differ by only a constant, which means they are equivalent up to an additive constant \( C \).
Step 2: Recall the trigonometric identity relating \( \sec(\theta) \) and \( \tan(\theta) \): \[ \sec^2(\theta) = 1 + \tan^2(\theta) \] Since \( \theta = \arctan x \), substitute to get: \[ \sec(\arctan x) = \sqrt{1 + x^2} \]
Step 3: Substitute \( \sec(\arctan x) \) in the first formula with \( \sqrt{1 + x^2} \) from the identity above. This shows that: \[ \ln \sec(\arctan x) = \ln \sqrt{1 + x^2} \]
Step 4: Recognize that the difference between the two integral expressions is just the constant of integration \( C \), since the logarithmic terms are equivalent by the identity. Therefore, both formulas are correct and represent the same family of antiderivatives.
Step 5: Conclude that different-looking expressions for an integral can be equivalent if they differ by a constant, which is always possible due to the arbitrary constant in indefinite integrals.

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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 is expressed as ∫u dv = uv - ∫v du. This method is often applied to integrals involving inverse trigonometric functions like arctan x.
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Integration by Parts for Definite Integrals

Inverse Trigonometric Functions and Their Properties

The arctan function is the inverse of the tangent function, mapping real numbers to angles. Understanding its properties, such as its range and how it relates to right triangles, helps simplify expressions involving arctan, especially when converting between trigonometric and algebraic forms.
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Derivatives of Other Inverse Trigonometric Functions

Logarithmic Identities and Simplification

Logarithmic expressions can often be rewritten using identities like ln(sec(arctan x)) = ln(√(1 + x²)). Recognizing these equivalences is crucial to show that different integral forms are actually the same up to a constant, confirming the correctness of both results.
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Verifying Trig Equations as Identities
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