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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 8.1.2
Chapter 8, Problem 8.1.2

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (x² / (x² + 1)) dx

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1
Start by rewriting the integrand to simplify the expression. Notice that \( \frac{x^2}{x^2 + 1} \) can be expressed as \( 1 - \frac{1}{x^2 + 1} \) because \( \frac{x^2}{x^2 + 1} = \frac{x^2 + 1 - 1}{x^2 + 1} = 1 - \frac{1}{x^2 + 1} \).
Split the integral into two separate integrals using the linearity of integration: \( \int \frac{x^2}{x^2 + 1} \, dx = \int 1 \, dx - \int \frac{1}{x^2 + 1} \, dx \).
Integrate the first integral \( \int 1 \, dx \), which is straightforward and equals \( x + C_1 \), where \( C_1 \) is a constant of integration.
Recognize that the second integral \( \int \frac{1}{x^2 + 1} \, dx \) is a standard integral that results in the inverse tangent function, \( \arctan(x) + C_2 \), where \( C_2 \) is another constant of integration.
Combine the results of both integrals to write the final expression as \( x - \arctan(x) + C \), where \( C = C_1 - C_2 \) is the overall constant of integration.

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Key Concepts

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

Integration by Algebraic Manipulation

This technique involves rewriting the integrand into a simpler form before integrating. For example, splitting a rational function into simpler terms can make the integral easier to evaluate without complex substitutions.
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Integration by Parts for Definite Integrals

Basic Integration Rules

Understanding fundamental integration formulas, such as the integral of powers of x and constants, is essential. These rules allow direct integration once the integrand is simplified.
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Basic Rules for Definite Integrals

Substitution Method

Substitution involves changing variables to simplify the integral, especially when the integrand contains composite functions. Identifying an inner function and its derivative helps transform the integral into a standard form.
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Euler's Method
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