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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 4.7.45
Chapter 4, Problem 4.7.45

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.


∫(sin2x − csc²x)dx

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1
Rewrite the integral by separating it into two simpler integrals: \(\int (\sin 2x - \csc^2 x) \, dx = \int \sin 2x \, dx - \int \csc^2 x \, dx\).
Recall the integral formulas for each term: the integral of \(\sin 2x\) and the integral of \(\csc^2 x\). Specifically, use the substitution method for \(\sin 2x\) and the known antiderivative for \(\csc^2 x\).
For \(\int \sin 2x \, dx\), let \(u = 2x\), so \(du = 2 \, dx\) or \(dx = \frac{du}{2}\). Rewrite the integral in terms of \(u\) and integrate.
For \(\int \csc^2 x \, dx\), recall that the derivative of \(\cot x\) is \(-\csc^2 x\), so the integral of \(\csc^2 x\) is \(-\cot x\) plus a constant.
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative. Finally, verify your answer by differentiating it to check if you get the original integrand.

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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 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

Integration of Trigonometric Functions

Integrating trigonometric functions like sin(2x) and csc²(x) requires knowledge of standard integral formulas and substitution techniques. For instance, ∫sin(2x)dx can be solved using a substitution for the inner function, while ∫csc²(x)dx is a standard integral equal to -cot(x) + C.
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Introduction to Trigonometric Functions

Verification by Differentiation

After finding an antiderivative, verifying the solution by differentiating it ensures correctness. Differentiation of the proposed integral should return the original integrand. This step confirms the integral was computed accurately and helps identify any errors.
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Related Practice
Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.


f(x) = (2/3)x − 5, −2 ≤ x ≤ 3

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Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

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Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dv/dt = (1/2)sec t tan t, v(0) = 1

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Textbook Question

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

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Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² − 32√x

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Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

2. y=x^4/4-2x^2+4

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