Evaluating Indefinite Integrals

Evaluate the integrals in Exercises 37–46.

∫ 2(cos x)⁻¹/² sin x dx

검증된 단계별 안내

Recognize that the integral is of the form \(\int 2 (\cos x)^{-\frac{1}{2}} \sin x \, dx\), which suggests a substitution involving \(\cos x\) because its derivative is related to \(\sin x\).

Let \(u = \cos x\). Then, compute the differential \(du = -\sin x \, dx\), which implies \(-du = \sin x \, dx\).

Rewrite the integral in terms of \(u\): substitute \(\cos x\) with \(u\) and \(\sin x \, dx\) with \(-du\), so the integral becomes \(\int 2 u^{-\frac{1}{2}} (-du)\).

Simplify the integral to \(-2 \int u^{-\frac{1}{2}} \, du\), which is a basic power integral.

Use the power rule for integration: \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\), and apply it to \(u^{-\frac{1}{2}}\) to find 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.

Indefinite Integrals

An indefinite integral represents the family of all antiderivatives of a function and is expressed without limits of integration. It includes a constant of integration (C) because differentiation of a constant is zero. Understanding indefinite integrals is essential for reversing differentiation and finding original functions.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable (u), rewriting the integral in terms of u, and then integrating. This technique is especially useful when the integral contains a function and its derivative.

Trigonometric Functions and Their Integrals

Trigonometric functions like sine and cosine have specific integral formulas and properties. Recognizing how to manipulate powers and compositions of trig functions is crucial. For example, integrating expressions involving powers of cosine or sine often requires substitution or trigonometric identities.

Evaluating Indefinite IntegralsEvaluate the integrals in Exercises 37–46.∫ 2(cos x)⁻¹/² sin x dx