Domanda del libro di testo
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (r² + r + 1) e^r dr
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Ch. 8 - Techniques of Integration
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 8, Problema 8.3.58
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
Guida verificata passo dopo passo1
Recognize that the integral involves the product of two cosine functions: \(\cos(x)\) and \(\cos(7x)\), integrated over the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Use the product-to-sum identity for cosine functions: \(\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]\). Applying this, rewrite the integrand as \(\frac{1}{2} [\cos(x - 7x) + \cos(x + 7x)] = \frac{1}{2} [\cos(-6x) + \cos(8x)]\).
Since cosine is an even function, \(\cos(-6x) = \cos(6x)\), so the integrand simplifies to \(\frac{1}{2} [\cos(6x) + \cos(8x)]\).
Split the integral into two separate integrals: \(\frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(6x) \, dx + \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(8x) \, dx\).
Evaluate each integral by finding the antiderivative of cosine, which is sine divided by the coefficient of \(x\), and then apply the limits of integration \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) for each term.
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Definite Integrals
A definite integral calculates the net area under a curve between two specified limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.
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Percorso guidato
05:43
Definition of the Definite Integral
Product-to-Sum Trigonometric Identities
These identities transform products of trigonometric functions into sums or differences, simplifying integration. For example, cos(A)cos(B) = ½[cos(A−B) + cos(A+B)], which helps break down complex integrals into easier terms.
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7:17Verifying Trig Equations as Identities
Symmetry of Trigonometric Functions
Understanding whether a function is even, odd, or neither helps simplify definite integrals over symmetric intervals. Cosine is an even function, so integrals from -a to a can be simplified by doubling the integral from 0 to a.
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6:04Introduction to Trigonometric Functions
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