Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫(from ln3 to ln2) (e^x) dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.2.49
Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt
Guida verificata passo dopo passo1
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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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.
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Substitution With an Extra Variable
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.
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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.
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6:36Simplifying Trig Expressions
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