The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds

검증된 단계별 안내

Identify the integral to be solved: \(\int_0^2 \frac{s + 1}{\sqrt{4 - s^2}} \, ds\).

Split the integral into two separate integrals for easier handling: \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds + \int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\).

For the first integral \(\int_0^2 \frac{s}{\sqrt{4 - s^2}} \, ds\), use the substitution method. Let \(u = 4 - s^2\), then find \(du\) and express \(s \, ds\) in terms of \(du\).

For the second integral \(\int_0^2 \frac{1}{\sqrt{4 - s^2}} \, ds\), recognize it as a standard integral form related to the arcsine function: \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C\).

Evaluate both integrals using the substitution and standard integral results, then combine the results and apply the limits from 0 to 2 to find the value of the original integral.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

도움이 되었나요?

주요 개념

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

Definite Integrals

A definite integral calculates the net area under a curve between two limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand equal to a new variable, which helps in integrating complex expressions like those involving square roots.

Integrals Involving Square Roots of Quadratic Expressions

Integrals with terms like √(a² − x²) often require trigonometric substitution or recognizing standard integral forms. These techniques help convert the integral into a trigonometric integral, which is easier to evaluate without tables.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀² (s + 1) / √(4 − s²) ds