Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))

검증된 단계별 안내

Start by examining the integral: \(\int \frac{dt}{t - \sqrt{1 - t^{2}}}\). Notice the expression in the denominator involves both \(t\) and \(\sqrt{1 - t^{2}}\), which suggests a trigonometric substitution might simplify the square root term.

Use the substitution \(t = \sin(\theta)\), which implies \(dt = \cos(\theta) d\theta\). This substitution is helpful because \(\sqrt{1 - t^{2}}\) becomes \(\sqrt{1 - \sin^{2}(\theta)} = \cos(\theta)\).

Rewrite the integral in terms of \(\theta\): replace \(t\) with \(\sin(\theta)\) and \(dt\) with \(\cos(\theta) d\theta\). The integral becomes \(\int \frac{\cos(\theta) d\theta}{\sin(\theta) - \cos(\theta)}\).

To simplify the integral, consider dividing numerator and denominator by \(\cos(\theta)\) (assuming \(\cos(\theta) \neq 0\)), which transforms the integral into \(\int \frac{d\theta}{\tan(\theta) - 1}\). This form is easier to handle.

Next, use the substitution \(u = \tan(\theta) - 1\), so that \(du = \sec^{2}(\theta) d\theta\). Express \(d\theta\) in terms of \(du\) and rewrite the integral accordingly. This will allow you to integrate with respect to \(u\) and then back-substitute to \(t\).

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주요 개념

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

Integration Techniques

Integration techniques involve methods such as substitution, partial fractions, or trigonometric substitution to simplify and evaluate integrals. Recognizing the form of the integrand helps determine the appropriate technique to apply for solving the integral efficiently.

Trigonometric Substitution

Trigonometric substitution is used to simplify integrals involving expressions like √(1 - t²) by substituting t = sin(θ) or t = cos(θ). This transforms the integral into a trigonometric form that is often easier to integrate.

Algebraic Manipulation of the Integrand

Algebraic manipulation involves rewriting the integrand to a simpler or more recognizable form, such as rationalizing the denominator or combining terms. This step is crucial to make the integral more approachable for standard integration methods.

Evaluate the integrals in Exercises 1–6.∫ dt / (t - √(1 - t²))