Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.8.32

Capitolo 8, Problema 8.8.32

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|

Guida verificata passo dopo passo

First, recognize that the integral involves the expression $\sqrt{|x - 1|}$, which has an absolute value inside the square root. This means the behavior of the integrand changes at $x = 1$, so we should split the integral at this point.

Rewrite the integral as the sum of two integrals: $\int_0^1 \frac{dx}{\sqrt{1 - x}} + \int_1^2 \frac{dx}{\sqrt{x - 1}}$. Notice how the absolute value affects the expression inside the square root on each interval.

For the first integral, $\int_0^1 \frac{dx}{\sqrt{1 - x}}$, use the substitution $u = 1 - x$, which implies $du = -dx$. Adjust the limits accordingly and rewrite the integral in terms of $u$.

For the second integral, $\int_1^2 \frac{dx}{\sqrt{x - 1}}$, use the substitution $v = x - 1$, which implies $dv = dx$. Adjust the limits accordingly and rewrite the integral in terms of $v$.

Evaluate both integrals using the power rule for integrals: $\int u^{n} du = \frac{u^{n+1}}{n+1} + C$ for $n \neq -1$. After evaluating, sum the results to get the value of the original integral.

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Improper Integrals

Improper integrals involve integrands with infinite discontinuities or infinite limits of integration. In this problem, the integrand has a singularity at x = 1, where the denominator becomes zero, requiring careful evaluation of the integral as a limit approaching the point of discontinuity.

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Percorso guidato

11:11

Improper Integrals: Infinite Intervals

Absolute Value and Piecewise Functions

The absolute value inside the square root splits the integral into separate intervals where the expression inside changes sign. Understanding how to rewrite |x - 1| as (1 - x) or (x - 1) depending on the interval is essential to correctly set up and evaluate the integral.

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Piecewise Functions

Integration Techniques for Power Functions

Integrating functions of the form 1/√(x - a) involves recognizing them as power functions with fractional exponents. Applying the power rule for integration, ∫x^n dx = x^(n+1)/(n+1), is necessary after rewriting the integrand appropriately to find the antiderivative.

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Representing Functions as Power Series

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