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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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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