Domanda del libro di testo
4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
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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.PE.135
135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:
(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.
Guida verificata passo dopo passo1
Start by defining the integral: \(I = \int_0^{\frac{\pi}{2}} \frac{\sin x}{\sin x + \cos x} \, dx\).
To evaluate the integral directly, consider the integrand \(f(x) = \frac{\sin x}{\sin x + \cos x}\). Find its antiderivative \(F(x)\) by using algebraic manipulation or substitution techniques.
One useful approach is to rewrite the integrand as \(\frac{\sin x}{\sin x + \cos x} = 1 - \frac{\cos x}{\sin x + \cos x}\), which might simplify the integration process.
Integrate each term separately: \(\int 1 \, dx\) and \(\int \frac{\cos x}{\sin x + \cos x} \, dx\). For the second integral, consider the substitution \(u = \sin x + \cos x\), then find \(du\) and express the integral in terms of \(u\).
After finding the antiderivative \(F(x)\), apply the Evaluation Theorem (Fundamental Theorem of Calculus) by computing \(F\left(\frac{\pi}{2}\right) - F(0)\) to get the value of the definite integral.
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Definite Integral and Evaluation Theorem
A definite integral calculates the net area under a curve between two limits. The Evaluation Theorem states that if F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) - F(a). This allows us to find the exact value of the integral by evaluating the antiderivative at the boundaries.
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Percorso guidato
05:43
Definition of the Definite Integral
Integration Techniques for Rational Trigonometric Functions
Integrating expressions like (sin x) / (sin x + cos x) often requires algebraic manipulation or substitution to simplify the integrand. Recognizing patterns or using substitutions such as t = tan x or rewriting numerator and denominator can make the integral more manageable.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Symmetry and Complementary Angle Properties in Definite Integrals
For integrals over [0, π/2], using symmetry or the property f(π/2 - x) can simplify evaluation. This approach exploits complementary angles in sine and cosine to relate integrals and sometimes find values without direct integration.
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Percorso guidato
05:43Definition of the Definite Integral
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Domanda del libro di testo
Heat capacity of a gas
Heat capacity
C_v
is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).
The heat capacity of oxygen depends on its temperature T and satisfies the formula
C_v = 8.27 + 10^(-5) * (26T − 1.87T²)
Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for
20°C ≤ T ≤ 675°C.
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Domanda del libro di testo
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
129. ∫ (x^(ln x) * ln x) / x dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx
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