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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.75
Capitolo 4, Problema 4.PE.75

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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∫ ( 3√ t + 4/t² ) dt

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Rewrite the integral by expressing the terms with exponents: \(\int \left( 3\sqrt{t} + \frac{4}{t^{2}} \right) dt = \int \left( 3t^{\frac{1}{2}} + 4t^{-2} \right) dt\).
Use the linearity of the integral to split it into two separate integrals: \(\int 3t^{\frac{1}{2}} dt + \int 4t^{-2} dt\).
Apply the power rule for integration to each term separately. Recall that for \(\int t^{n} dt\), the antiderivative is \(\frac{t^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Integrate the first term: \(\int 3t^{\frac{1}{2}} dt = 3 \cdot \frac{t^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = 3 \cdot \frac{t^{\frac{3}{2}}}{\frac{3}{2}}\).
Integrate the second term: \(\int 4t^{-2} dt = 4 \cdot \frac{t^{-2 + 1}}{-2 + 1} = 4 \cdot \frac{t^{-1}}{-1}\), then combine both results and add the constant of integration \(+ C\).

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Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed as a family of functions plus a constant of integration (C). It reverses differentiation and is written without limits, indicating all possible antiderivatives.
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that ∫ t^n dt = (t^(n+1)) / (n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like t^(1/3) or t^(-2), allowing us to find antiderivatives by increasing the exponent by one and dividing by the new exponent.
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Power Rule for Indefinite Integrals

Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors in the integration process.
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Finding Differentials
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Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

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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.

∫ cos³ 𝓍/2 d𝓍

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Domanda del libro di testo

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

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Domanda del libro di testo

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 + 5𝓍 ―7) d𝓍

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Domanda del libro di testo

Identifying Extrema


In Exercises 15–18:


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


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


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Domanda del libro di testo

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.

∫ sec θ/3 tan θ/3 dθ

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