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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 4.7.51
Chapter 4, Problem 4.7.51

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.


∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)

Verified step by step guidance
1
Recognize the integral given: \(\int (1 + \tan^{2}\theta) \, d\theta\).
Use the hint provided, which states that \(1 + \tan^{2}\theta = \sec^{2}\theta\). Substitute this into the integral to simplify it to \(\int \sec^{2}\theta \, d\theta\).
Recall the basic integral formula: \(\int \sec^{2}x \, dx = \tan x + C\), where \(C\) is the constant of integration.
Apply this formula to the integral, replacing \(x\) with \(\theta\), so the antiderivative is \(\tan \theta + C\).
Verify your result by differentiating \(\tan \theta + C\) to check if it returns the original integrand \(1 + \tan^{2}\theta\).

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Key Concepts

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

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It is written without limits and describes a family of functions whose derivative is the integrand.
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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the identity 1 + tan²θ = sec²θ simplifies integrals by rewriting expressions into more manageable forms.
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Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the indefinite integral and ensures no errors were made during integration.
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Related Practice
Textbook Question

41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?

Textbook Question

22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

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Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = x⁴ᐟ⁵, [0, 1]

Textbook Question

Find values of a and b such that the function


ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)


has a local extreme value of 1 at 𝓍 = 3. Is this extreme value a local maximum or a local minimum? Give reasons for your answer.

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 3x⁻²ᐟ³, y(−1) = −5

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.

4. y=9/14x^(1/3)(x^2-7)