Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt

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Identify the integral to solve: \(\int \frac{3 \sec^{2} t}{6 + 3 \tan t} \, dt\).

Notice that the denominator contains \(6 + 3 \tan t\), and the numerator contains \(3 \sec^{2} t\). Recall that the derivative of \(\tan t\) is \(\sec^{2} t\), which suggests a substitution.

Let \(u = 6 + 3 \tan t\). Then, compute \(du\): since \(\frac{d}{dt}(\tan t) = \sec^{2} t\), we have \(du = 3 \sec^{2} t \, dt\).

Rewrite the integral in terms of \(u\) and \(du\): the numerator \(3 \sec^{2} t \, dt\) becomes \(du\), and the denominator is \(u\), so the integral becomes \(\int \frac{1}{u} \, du\).

Integrate \(\int \frac{1}{u} \, du\) to get \(\ln|u| + C\), then substitute back \(u = 6 + 3 \tan t\) to express the answer in terms of \(t\).

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주요 개념

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Integration Using Substitution

Substitution is a method to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative also appears in the integral, allowing the integral to be rewritten in terms of a new variable. This technique is especially useful when the integral contains composite functions like trigonometric expressions.

Trigonometric Identities and Derivatives

Understanding the derivatives and identities of trigonometric functions such as tan(t) and sec²(t) is crucial. For example, the derivative of tan(t) is sec²(t), which helps in recognizing substitution candidates. Familiarity with these relationships simplifies the integration process involving trigonometric functions.

Simplifying Rational Expressions in Integrals

When integrating rational expressions, simplifying the numerator and denominator can make the integral more manageable. Factoring constants and rewriting expressions help in identifying substitution opportunities and reducing the integral to a standard form that is easier to evaluate.

Evaluate the integrals in Exercises 39–56.49. ∫3sec²t/(6 + 3tan(t)) dt