Evaluate the integrals in Exercises 91–102.
102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx

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1. Recognize that the integral involves the expression \( \cos(\arctan(3x)) \) and the denominator \( 1 + 9x^2 \). This suggests a trigonometric substitution related to the angle \( \theta = \arctan(3x) \).
2. Set \( \theta = \arctan(3x) \), which implies \( 3x = \tan(\theta) \). From this, express \( x \) in terms of \( \theta \): \( x = \frac{\tan(\theta)}{3} \).
3. Calculate the differential \( dx \) in terms of \( d\theta \). Since \( x = \frac{\tan(\theta)}{3} \), then \( dx = \frac{1}{3} \sec^2(\theta) d\theta \).
4. Rewrite the integral limits in terms of \( \theta \) by substituting the original \( x \) limits into \( \theta = \arctan(3x) \). For \( x = -\frac{1}{3} \), \( \theta = \arctan(-1) = -\frac{\pi}{4} \). For \( x = \frac{1}{\sqrt{3}} \), \( \theta = \arctan\left(3 \cdot \frac{1}{\sqrt{3}}\right) = \arctan(\sqrt{3}) = \frac{\pi}{3} \).
5. Substitute all parts into the integral: replace \( \cos(\arctan(3x)) \) with \( \cos(\theta) \), replace \( 1 + 9x^2 \) with \( 1 + \tan^2(\theta) = \sec^2(\theta) \), and replace \( dx \) with \( \frac{1}{3} \sec^2(\theta) d\theta \). Simplify the integrand and integral accordingly before integrating with respect to \( \theta \) from \( -\frac{\pi}{4} \) to \( \frac{\pi}{3} \).

Key Concepts

Integration of Composite Functions

This involves integrating functions composed of other functions, such as trigonometric functions of inverse trigonometric functions. Recognizing the inner function and its derivative helps in applying substitution methods to simplify the integral.

Inverse Trigonometric Functions and Their Properties

Understanding the definition and properties of inverse trigonometric functions like arctan is crucial. For example, arctan(x) returns an angle whose tangent is x, which allows rewriting expressions involving arctan in terms of trigonometric identities.

Substitution Method in Definite Integrals

The substitution method replaces a complicated expression with a single variable to simplify integration. When dealing with definite integrals, the limits must be adjusted according to the substitution to correctly evaluate the integral.