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 8.6.56

Capitolo 8, Problema 8.6.56

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

Guida verificata passo dopo passo

Recognize that the integral has the form \(\int \frac{dt}{(t^2 + 1)^{7/2}}\), which suggests a trigonometric substitution because of the \(t^2 + 1\) term inside the power.

Use the substitution \(t = \tan(\theta)\), which implies \(dt = \sec^2(\theta) d\theta\) and \(t^2 + 1 = \sec^2(\theta)\).

Rewrite the integral in terms of \(\theta\): replace \(dt\) with \(\sec^2(\theta) d\theta\) and \((t^2 + 1)^{7/2}\) with \((\sec^2(\theta))^{7/2} = \sec^7(\theta)\), so the integrand becomes \(\frac{\sec^2(\theta)}{\sec^7(\theta)} = \sec^{-5}(\theta)\).

Change the limits of integration from \(t\) to \(\theta\): when \(t=0\), \(\theta = \arctan(0) = 0\); when \(t=\frac{1}{\sqrt{3}}\), \(\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\).

Now the integral is \(\int_0^{\pi/6} \sec^{-5}(\theta) d\theta = \int_0^{\pi/6} \cos^5(\theta) d\theta\). Use a reduction formula for powers of cosine to evaluate this integral step-by-step.

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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a tan θ), the integral transforms into a trigonometric integral that is often easier to evaluate.

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Introduction to Trigonometric Functions

Reduction Formulas

Reduction formulas express an integral with a certain power or parameter in terms of a similar integral with a lower power or simpler parameter. They help solve complex integrals step-by-step by breaking them down into simpler, more manageable integrals.

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Recursive Formulas

Definite Integration with Limits

Definite integration involves evaluating the integral between specified limits. When using substitution, it is important to change the limits according to the substitution or revert to the original variable before applying the limits to find the exact numerical value.

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Definition of the Definite Integral

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