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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.25
Chapter 4, Problem 4.7.25

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫x⁻¹ᐟ³ dx

Verified step by step guidance
1
Recognize that the integral is of the form \(\int x^{m} \, dx\) where the exponent \(m\) is \(-\frac{1}{3}\).
Recall the power rule for integration: for any \(m \neq -1\), \(\int x^{m} \, dx = \frac{x^{m+1}}{m+1} + C\) where \(C\) is the constant of integration.
Calculate the new exponent by adding 1 to \(m\): \(m + 1 = -\frac{1}{3} + 1 = \frac{2}{3}\).
Apply the power rule formula: \(\int x^{-\frac{1}{3}} \, dx = \frac{x^{\frac{2}{3}}}{\frac{2}{3}} + C\).
Simplify the fraction in the denominator by multiplying numerator and denominator appropriately, and remember to add the constant of integration \(C\).

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Key Concepts

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

Indefinite Integral and Antiderivative

An indefinite integral represents the most general form of the antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(x) dx = F(x) + C, where F'(x) = f(x).
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1. This rule is essential for integrating functions with variable exponents, such as x raised to fractional powers, by increasing the exponent by one and dividing by the new exponent.
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Power Rule for Indefinite Integrals

Handling Fractional and Negative Exponents

When integrating expressions like x raised to a fractional or negative exponent, treat the exponent as a rational number. Apply the power rule carefully, ensuring the exponent is not -1, and simplify the result. For example, x^(-1/3) integrates to (x^(2/3))/(2/3) + C.
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Zero and Negative Rules
Related Practice
Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dr/dθ = −π sin (πθ), r(0) = 0

Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.


g(θ) = θ³ᐟ⁵, −32 ≤ θ ≤ 1

Textbook Question

Motion with constant acceleration The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is s = (a/2)t² + v₀t + s₀, where v₀ and s₀ are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem

Differential equation: d²s/dt² = a

Initial conditions: ds/dt = v₀ and s = s₀ when t=0.

Textbook Question

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


f(x) =√(x − 1), [1, 3]

Textbook Question

Solve the initial value problems in Exercises 71–90.

y⁽⁴⁾ = −sin t + cos t;

y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0

Textbook Question

Business and Economics

60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.