Domanda del libro di testo
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.4.65
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
Guida verificata passo dopo passo1
Start by examining the quadratic expression in the denominator: \(8x^{2} - 8x + 11\). To simplify the integral, we want to complete the square for this quadratic.
Factor out the coefficient of \(x^{2}\) from the first two terms: \(8(x^{2} - x) + 11\).
Complete the square inside the parentheses: take half of the coefficient of \(x\), which is \(-1\), divide by 2 to get \(-\frac{1}{2}\), then square it to get \(\left(-\frac{1}{2}\right)^{2} = \frac{1}{4}\). Add and subtract this inside the parentheses:
\(8\left(x^{2} - x + \frac{1}{4} - \frac{1}{4}\right) + 11 = 8\left(\left(x - \frac{1}{2}\right)^{2} - \frac{1}{4}\right) + 11\).
Distribute the 8 and simplify the constant terms: \(8\left(x - \frac{1}{2}\right)^{2} - 8 \times \frac{1}{4} + 11 = 8\left(x - \frac{1}{2}\right)^{2} - 2 + 11 = 8\left(x - \frac{1}{2}\right)^{2} + 9\).
Rewrite the integral using this completed square form: \(\int_{\frac{1}{2}}^{\frac{\sqrt{2} + 3}{2\sqrt{2}}} \frac{dx}{8\left(x - \frac{1}{2}\right)^{2} + 9}\). Next, factor out the 9 to express the denominator in a form suitable for an arctangent substitution.
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Completing the Square
Completing the square is a technique used to rewrite quadratic expressions in the form ax² + bx + c as a perfect square plus or minus a constant. This simplifies integration by transforming the denominator into a form that matches standard integral formulas, especially those involving inverse trigonometric functions.
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05:22
Completing the Square to Rewrite the Integrand
Integration of Rational Functions
Integrating rational functions often involves algebraic manipulation such as factoring or completing the square. Recognizing the form of the denominator helps in applying appropriate substitution or standard integral results, enabling the evaluation of integrals that might otherwise be complex.
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6:04Intro to Rational Functions
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two specific points. Understanding how to apply the limits after finding the antiderivative is crucial, as it involves substituting the upper and lower bounds correctly to find the exact value of the integral.
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Percorso guidato
05:43Definition of the Definite Integral
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Domanda del libro di testo
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
18. ∫ dx / (225 − 16x²)
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Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
14. ∫ s · e⁻²ˢ ds
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Domanda del libro di testo
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx
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Domanda del libro di testo
4. Is a reduction formula an analytical method or a numerical method? Explain.
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Domanda del libro di testo
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
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