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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.4.20
Chapter 8, Problem 8.4.20

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (4 + x²)

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Start by examining the integral \( \int \frac{x^{2}}{4 + x^{2}} \, dx \). Notice that the denominator is \(4 + x^{2}\), which suggests a trigonometric substitution might be useful, but first consider simplifying the integrand by algebraic manipulation.
Rewrite the integrand by expressing \(x^{2}\) in terms of \(4 + x^{2}\): \(x^{2} = (4 + x^{2}) - 4\). Substitute this into the integral to get \(\int \frac{(4 + x^{2}) - 4}{4 + x^{2}} \, dx = \int \left(1 - \frac{4}{4 + x^{2}}\right) \, dx\).
Split the integral into two simpler integrals: \(\int 1 \, dx - 4 \int \frac{1}{4 + x^{2}} \, dx\). This separates the problem into a basic integral and a standard form integral involving \(\frac{1}{a^{2} + x^{2}}\).
Recall the standard integral formula: \(\int \frac{1}{a^{2} + x^{2}} \, dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C\). Here, \(a^{2} = 4\), so \(a = 2\). Use this to write the second integral explicitly.
Combine the results: the integral becomes \(\int 1 \, dx - 4 \times \frac{1}{2} \arctan \left( \frac{x}{2} \right) + C\). Simplify the constants and write the final expression in terms of \(x\) and \(\arctan\).

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Key Concepts

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

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like a² + x², a² - x², or x² - a² by substituting x with a trigonometric function. This transforms the integral into a trigonometric integral, which is often easier to evaluate. For example, substituting x = 2 tan(θ) can simplify integrals with 4 + x².
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Polynomial Division in Integration

When the degree of the numerator is equal to or greater than the denominator, polynomial division can simplify the integrand. Dividing the numerator by the denominator breaks the integral into simpler parts, often a polynomial plus a proper rational function, making the integral easier to solve.
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Basic Integration Techniques

Understanding fundamental integration methods such as substitution, recognizing standard integral forms, and using algebraic manipulation is essential. These techniques help in breaking down complex integrals into manageable parts and applying known formulas or substitutions effectively.
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Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of ((x dx) / √(x⁴ - 1))

Textbook Question

Solve the initial value problems in Exercises 53–56 for y as a function of x.

(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1

Textbook Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 8 cot^4(t) dt

Textbook Question

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (y + 4) / (y² + y) dy from 1/2 to 1

Textbook Question

87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.

Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / √(1 - x²)