Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)

검증된 단계별 안내

Recognize the integral given: \(\int (1 + \tan^{2}\theta) \, d\theta\).

Use the hint provided, which states that \(1 + \tan^{2}\theta = \sec^{2}\theta\). Substitute this into the integral to simplify it to \(\int \sec^{2}\theta \, d\theta\).

Recall the basic integral formula: \(\int \sec^{2}x \, dx = \tan x + C\), where \(C\) is the constant of integration.

Apply this formula to the integral, replacing \(x\) with \(\theta\), so the antiderivative is \(\tan \theta + C\).

Verify your result by differentiating \(\tan \theta + C\) to check if it returns the original integrand \(1 + \tan^{2}\theta\).

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영상 길이:

45s

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주요 개념

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

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It is written without limits and describes a family of functions whose derivative is the integrand.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the identity 1 + tan²θ = sec²θ simplifies integrals by rewriting expressions into more manageable forms.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the indefinite integral and ensures no errors were made during integration.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)