Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.2.40

Capitolo 8, Problema 8.2.40

9–40. Integration by parts Evaluate the following integrals using integration by parts.
40. ∫ e^√x dx

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Identify the integral to solve: $\int e^{\sqrt{x}} \, dx$.

Use substitution to simplify the integral. Let $t = \sqrt{x}$, which implies $x = t^2$. Then, differentiate to find $dx$: $dx = 2t \, dt$.

Rewrite the integral in terms of $t$: $\int e^{t} \, dx = \int e^{t} \cdot 2t \, dt = 2 \int t e^{t} \, dt$.

Apply integration by parts to $\int t e^{t} \, dt$. Choose $u = t$ (so $du = dt$) and $dv = e^{t} dt$ (so $v = e^{t}$). Use the formula $\int u \, dv = uv - \int v \, du$.

Substitute back the expressions for $u$, $v$, and $du$ into the integration by parts formula to express the integral, then multiply by 2 as per the substitution step. Finally, replace $t$ with $\sqrt{x}$ to return to the original variable.

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Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.

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Integration by Parts for Definite Integrals

Substitution Method

The substitution method involves changing variables to simplify an integral. For integrals like ∫e^√x dx, substituting t = √x helps rewrite the integral in terms of t, making it easier to apply integration techniques such as integration by parts.

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Euler's Method

Handling Composite Functions in Integration

Integrating functions like e^√x requires understanding how to deal with composite functions, where one function is nested inside another. Recognizing the inner function and its derivative is essential for applying substitution or integration by parts correctly.

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Evaluate Composite Functions - Special Cases

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76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

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64. (Use of Tech) Normal distribution of movie lengths

A study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integral:

(1/(22√(2π))) ∫[a to b] e^(-((x-110)/22)²/2) dx.

What percentage of U.S. movies are between 1 hr and 1.5 hr long (60-90 min)?

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7–84. Evaluate the following integrals.

25. ∫ [1 / (x√(1 - x²))] dx

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7–58. Improper integrals Evaluate the following integrals or state that they diverge.

50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx

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