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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.PE.81
Capitolo 4, Problema 4.PE.81

Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

Guida verificata passo dopo passo
1
Identify the integral to solve: \(\int x^{3} (1 + x^{4})^{-\frac{1}{4}} \, dx\).
Look for a substitution that simplifies the integral. Notice that the expression inside the parentheses is \(1 + x^{4}\), and its derivative involves \(x^{3}\). This suggests using the substitution \(u = 1 + x^{4}\).
Compute the differential \(du\) by differentiating \(u\) with respect to \(x\): \(du = 4x^{3} \, dx\). From this, solve for \(x^{3} \, dx\) to express it in terms of \(du\): \(x^{3} \, dx = \frac{du}{4}\).
Rewrite the integral in terms of \(u\) using the substitution: replace \(x^{3} \, dx\) with \(\frac{du}{4}\) and \((1 + x^{4})^{-\frac{1}{4}}\) with \(u^{-\frac{1}{4}}\). The integral becomes \(\int u^{-\frac{1}{4}} \cdot \frac{du}{4}\).
Integrate the new integral with respect to \(u\): \(\frac{1}{4} \int u^{-\frac{1}{4}} \, du\). Use the power rule for integration, which states \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\). After integrating, substitute back \(u = 1 + x^{4}\) to express the answer in terms of \(x\).

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Indefinite Integrals and Antiderivatives

An indefinite integral represents the most general antiderivative of a function, including an arbitrary constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this concept is essential for solving integrals and verifying solutions by differentiation.
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Substitution Method

The substitution method simplifies integration by changing variables to transform a complex integral into a more manageable form. Typically, a part of the integrand is set as a new variable, allowing the integral to be rewritten in terms of this variable. This technique is especially useful when the integrand contains composite functions.
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Power Rule for Integration

The power rule states that the integral of x^n (for n ≠ -1) is (x^(n+1))/(n+1) plus a constant. This rule is fundamental for integrating polynomial expressions and powers of variables. Recognizing when to apply this rule after substitution helps in efficiently solving integrals involving powers.
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Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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Absolute Extrema on Finite Closed Intervals


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Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


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