29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀³ (2x - 1) / (x + 1) dx

검증된 단계별 안내

Start by examining the integral \( \int_0^3 \frac{2x - 1}{x + 1} \, dx \). Notice that the integrand is a rational function where the degree of the numerator is equal to the degree of the denominator.

Perform polynomial division to simplify the integrand. Divide \( 2x - 1 \) by \( x + 1 \) to express the integrand as a polynomial plus a proper fraction. This will help in integrating more easily.

After division, rewrite the integral as \( \int_0^3 \left( \text{quotient} + \frac{\text{remainder}}{x + 1} \right) dx \). This separates the integral into simpler parts.

Integrate each part separately: the polynomial part integrates to a power function, and the fraction part integrates to a logarithmic function involving \( \ln|x + 1| \). Since \( x + 1 > 0 \) on \([0,3]\), absolute value is not necessary here.

Finally, apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits \( x=3 \) and \( x=0 \), then subtract to find the value of the definite 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 algebraic manipulation such as polynomial division or partial fraction decomposition to simplify the integrand before integrating.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specified points. The limits of integration (here, 0 to 3) define the interval over which the function is integrated, resulting in a numerical value.

Use of Absolute Values in Logarithmic Integrals

When integrating functions that lead to logarithmic expressions, absolute values ensure the argument of the logarithm remains positive. Absolute values are included only when the domain of the function requires it to maintain validity.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₀³ (2x - 1) / (x + 1) dx