58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.

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Start with the integral \( \int_0^{\frac{\pi}{3}} \sin(x) \cdot \ln(\cos(x)) \, dx \). The goal is to use substitution to simplify this integral.

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

Rewrite the integral in terms of \( u \). When \( x = 0 \), \( u = \cos(0) = 1 \). When \( x = \frac{\pi}{3} \), \( u = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). Substitute these limits and the expression for \( \sin(x) \, dx \) into the integral:

\[ \int_0^{\frac{\pi}{3}} \sin(x) \ln(\cos(x)) \, dx = \int_1^{\frac{1}{2}} \ln(u) (-du) = \int_{\frac{1}{2}}^{1} \ln(u) \, du \]

Now, the integral is \( \int_{\frac{1}{2}}^{1} \ln(u) \, du \), which can be evaluated using integration by parts or known formulas for \( \int \ln(u) \, du \).

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주요 개념

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Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a more manageable form, often by letting a part of the integrand equal a new variable. This method is especially useful when the integral contains a composite function.

Properties of Logarithmic and Trigonometric Functions

Understanding the behavior and derivatives of logarithmic and trigonometric functions is essential. For example, knowing that the derivative of ln(cos(x)) involves -tan(x) helps in choosing an effective substitution. Familiarity with these functions aids in manipulating the integral and recognizing suitable substitutions.

Definite Integrals and Limits of Integration

When performing substitution in definite integrals, it is important to adjust the limits of integration according to the new variable. This avoids the need to revert to the original variable after integration and ensures the integral is evaluated correctly over the specified interval.

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:b. Use substitution.