2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

검증된 단계별 안내

Step 1: Recognize that the denominator of the integrand, 4x² + 12x + 10, is a quadratic expression. To simplify the integral, complete the square for the quadratic expression in the denominator.

Step 2: Rewrite 4x² + 12x + 10 by factoring out 4 from the first two terms: 4(x² + 3x) + 10. Then complete the square inside the parentheses: x² + 3x = (x + 3/2)² - (3/2)². Substitute this back into the expression.

Step 3: After completing the square, the denominator becomes 4((x + 3/2)² - 9/4) + 10. Simplify this expression to get 4(x + 3/2)² + 1.

Step 4: Recognize that the integral now has the form ∫ dx / [a² + (x - h)²], which is a standard form for integration using the arctangent function. Use the formula ∫ dx / (a² + x²) = (1/a) * arctan(x/a) + C, adjusting for the shift in x.

Step 5: Apply the limits of integration (-3/2 to -1) to the result obtained from the arctangent formula. Evaluate the definite integral by substituting the limits into the antiderivative.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is essential for evaluating more complex integrals, especially those that cannot be solved using basic antiderivatives.

Definite Integrals

A definite integral calculates the area under a curve between two specified limits. It is represented as ∫ from a to b f(x) dx, where 'a' and 'b' are the bounds of integration. The result of a definite integral is a numerical value that represents this area, which is crucial for applications in physics and engineering.

Quadratic Functions

Quadratic functions are polynomial functions of the form ax² + bx + c, where a, b, and c are constants. In the context of integration, recognizing the structure of a quadratic expression can help in simplifying the integral, often by completing the square or using substitution to facilitate easier integration.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)