Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)

검증된 단계별 안내

Identify the integral to evaluate: \(\displaystyle \int_0^{\pi} \frac{\cos(x)}{\sqrt{1 + \sin^2(x)}} \, dx\).

Recognize that the integrand contains \(\cos(x)\) and \(\sin(x)\), suggesting a substitution involving \(\sin(x)\) might simplify the integral.

Use the substitution \(u = \sin(x)\), which implies \(du = \cos(x) \, dx\). This transforms the integral into \(\int_{u=\sin(0)}^{u=\sin(\pi)} \frac{1}{\sqrt{1 + u^2}} \, du\).

Evaluate the new limits: since \(\sin(0) = 0\) and \(\sin(\pi) = 0\), the integral becomes \(\int_0^0 \frac{1}{\sqrt{1 + u^2}} \, du\).

Notice that the limits are the same, so the integral evaluates to zero. However, consider the behavior of the function on the interval and whether the substitution covers the entire integral correctly, possibly splitting the integral at \(\pi/2\) to handle the sign of \(\cos(x)\).

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주요 개념

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful when the integrand contains a composite function, allowing the integral to be expressed in terms of a new variable, often leading to easier integration.

Natural Logarithms in Integration

Natural logarithms often appear when integrating functions of the form f'(x)/f(x), resulting in ln|f(x)| + C. Recognizing when an integral can be expressed in terms of natural logarithms helps simplify the evaluation, especially when the integrand involves expressions under a root or rational functions.

Evaluate the integrals in Exercises 67–74 in terms ofb. natural logarithms.73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)