Evaluate the integrals in Exercises 77–90.
87. ∫(x²+2x-1)/(x²+9) dx

검증된 단계별 안내

Start by rewriting the integral: \(\int \frac{x^{2} + 2x - 1}{x^{2} + 9} \, dx\).

Split the integral into separate parts by expressing the numerator in a form related to the denominator: write \(x^{2} + 2x - 1\) as \((x^{2} + 9) + (2x - 10)\), so the integral becomes \(\int \frac{x^{2} + 9}{x^{2} + 9} \, dx + \int \frac{2x - 10}{x^{2} + 9} \, dx\).

Simplify the first integral to \(\int 1 \, dx\), and keep the second integral as \(\int \frac{2x - 10}{x^{2} + 9} \, dx\).

For the second integral, split it into two integrals: \(\int \frac{2x}{x^{2} + 9} \, dx - \int \frac{10}{x^{2} + 9} \, dx\). Use substitution for the first part and recognize the standard form for the second part.

Use substitution \(u = x^{2} + 9\) for \(\int \frac{2x}{x^{2} + 9} \, dx\), and recall that \(\int \frac{1}{x^{2} + a^{2}} \, dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C\) for the second integral.

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주요 개념

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

Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include polynomial division, partial fraction decomposition, or algebraic manipulation to simplify the integrand before integration.

Polynomial Division

When the degree of the numerator is equal to or greater than the denominator, polynomial division helps rewrite the integrand as a polynomial plus a proper fraction, simplifying the integral into more manageable parts.

Standard Integral Forms Involving Quadratics

Integrals with denominators like x² + a² often lead to inverse trigonometric functions such as arctangent. Recognizing these forms allows direct application of known integral formulas to solve the problem efficiently.

Evaluate the integrals in Exercises 77–90.87. ∫(x²+2x-1)/(x²+9) dx