Ch. 5 - Integrals

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 5 - Integrals

Problem 5.PE.56

Chapter 5, Problem 5.PE.56

Evaluate the integrals in Exercises 47–68.

∫₀¹/² x³ (1 + 9x⁴)⁻³/² dx

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Identify the integral to solve: \(\int_0^{\frac{1}{2}} x^3 (1 + 9x^4)^{-\frac{3}{2}} \, dx\).

Look for a substitution that simplifies the expression inside the integral. Notice that the term \(1 + 9x^4\) is raised to a power and \(x^3\) is multiplied outside. Consider letting \(u = 1 + 9x^4\).

Compute the differential \(du\) by differentiating \(u\) with respect to \(x\): \(du = 36x^3 \, dx\). This allows you to express \(x^3 \, dx\) in terms of \(du\) as \(x^3 \, dx = \frac{du}{36}\).

Change the limits of integration from \(x\) to \(u\): when \(x=0\), \(u = 1 + 9(0)^4 = 1\); when \(x=\frac{1}{2}\), \(u = 1 + 9\left(\frac{1}{2}\right)^4 = 1 + 9 \times \frac{1}{16} = 1 + \frac{9}{16} = \frac{25}{16}\).

Rewrite the integral in terms of \(u\) using the substitution and new limits: \(\int_1^{\frac{25}{16}} u^{-\frac{3}{2}} \times \frac{1}{36} \, du\). Then, proceed to integrate \(u^{-\frac{3}{2}}\) with respect to \(u\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values. This process provides a numerical value representing the accumulation of quantities over the interval.

Substitution Method

The substitution method simplifies integrals by changing variables to transform a complex integral into a more manageable form. By setting a part of the integrand as a new variable, the integral often becomes easier to evaluate, especially when dealing with composite functions.

Power Functions and Exponents

Understanding how to manipulate power functions and exponents is essential for integrating expressions involving terms like x³ and (1 + 9x⁴)⁻³/². This includes applying exponent rules and recognizing how powers affect differentiation and integration.

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