In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ dt / (tan(t)√4 - sin^2(t))

검증된 단계별 안내

Identify the integral: \(\int \frac{dt}{\tan(t) \sqrt{4 - \sin^2(t)}}\).

Rewrite the integral in terms of sine and cosine to simplify: recall that \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), so the integral becomes \(\int \frac{dt}{\frac{\sin(t)}{\cos(t)} \sqrt{4 - \sin^2(t)}} = \int \frac{\cos(t)}{\sin(t) \sqrt{4 - \sin^2(t)}} dt\).

Use the substitution \(u = \sin(t)\), which implies \(du = \cos(t) dt\). This substitution will help rewrite the integral in terms of \(u\).

Rewrite the integral in terms of \(u\): since \(\cos(t) dt = du\), the integral becomes \(\int \frac{du}{u \sqrt{4 - u^2}}\).

Recognize that the integral \(\int \frac{du}{u \sqrt{4 - u^2}}\) is a standard form that can be found in integral tables or solved using further substitution or partial fractions.

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.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of expressions like a² - x², a² + x², or x² - a². By substituting a trigonometric function for the variable, the integral transforms into a form involving trigonometric identities, making it easier to evaluate.

Integration Using Tables

Integration tables provide standard integral forms and their solutions, allowing quick evaluation of complex integrals once transformed appropriately. After substitution, matching the integral to a known form in the table helps in directly writing down the antiderivative without performing the integration from scratch.

Manipulation of Trigonometric Functions

Understanding the relationships and identities between trigonometric functions like sine, cosine, and tangent is essential. Simplifying expressions such as tan(t) and √(4 - sin²(t)) often requires using identities like sin²(t) + cos²(t) = 1, which aids in rewriting the integral into a more manageable form.

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ dt / (tan(t)√4 - sin^2(t))