Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ 6 dt / (9t² + 1)²

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Recognize that the integral has the form \(\int \frac{6}{(9t^{2} + 1)^{2}} \, dt\), which suggests a trigonometric substitution because of the quadratic expression inside the denominator.

Rewrite the denominator as \(9t^{2} + 1 = (3t)^{2} + 1^{2}\), which matches the form \(a^{2} + x^{2}\) where \(a = 1\) and \(x = 3t\).

Use the substitution \(3t = \tan(\theta)\), which implies \(t = \frac{1}{3} \tan(\theta)\) and \(dt = \frac{1}{3} \sec^{2}(\theta) \, d\theta\).

Rewrite the integral in terms of \(\theta\) by substituting \(t\) and \(dt\), and simplify the expression using the identity \(1 + \tan^{2}(\theta) = \sec^{2}(\theta)\).

Evaluate the resulting integral in \(\theta\), then substitute back \(\theta = \arctan(3t)\) to express the answer in terms of \(t\).

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주요 개념

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like a² + x², a² - x², or x² - a² by substituting x with a trigonometric function. This transforms the integral into a trigonometric integral that is often easier to evaluate. For example, substituting t = (1/3)tan(θ) can simplify the denominator 9t² + 1.

Integration of Rational Functions

Integrals involving rational functions, especially those with quadratic expressions in the denominator, often require algebraic manipulation or substitution. Recognizing the form and applying appropriate methods like partial fractions or trigonometric substitution helps in evaluating these integrals effectively.

Chain Rule in Reverse (Substitution Method)

The substitution method involves changing variables to simplify an integral, effectively reversing the chain rule from differentiation. By choosing a substitution that simplifies the integral's expression, such as setting u = 9t² + 1, the integral becomes easier to solve.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ 6 dt / (9t² + 1)²