Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)

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Recognize that the integral is of the form \(\int \frac{dx}{a^2 + (x - b)^2}\), which is a standard integral related to the arctangent function.

Rewrite the integral to clearly identify \(a\) and \(b\): here, \(a^2 = 2\) so \(a = \sqrt{2}\), and \(b = 1\).

Recall the formula for the integral: \(\int \frac{dx}{a^2 + (x - b)^2} = \frac{1}{a} \arctan\left( \frac{x - b}{a} \right) + C\).

Apply the formula by substituting \(a = \sqrt{2}\) and \(b = 1\) into the expression to write the antiderivative.

Include the constant of integration \(C\) to complete the indefinite 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 ratios of polynomials or expressions that can be manipulated into such forms. Recognizing the structure of the integrand helps in applying appropriate techniques like substitution or partial fractions.

Integration of Functions Involving Quadratic Expressions

When the integrand contains a quadratic expression in the denominator, completing the square or recognizing standard integral forms is essential. This often leads to integrals resembling the arctangent function.

Arctangent Integral Formula

The integral of 1/(a² + x²) dx is (1/a) arctan(x/a) + C. This formula is key for integrals where the denominator is a sum of a constant squared and a squared variable term, allowing direct evaluation using inverse trigonometric functions.

Evaluate the integrals in Exercises 53–76.67. ∫dx/(2+(x-1)²)