29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₁ᵉ^² (ln x)^5 / x dx

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Identify the integral to solve: \(\int_1^{e^2} \frac{(\ln x)^5}{x} \, dx\).

Recognize that the integrand contains \((\ln x)^5\) divided by \(x\), which suggests using the substitution \(u = \ln x\).

Compute the differential \(du\): since \(u = \ln x\), then \(du = \frac{1}{x} dx\), which matches the \(\frac{1}{x} dx\) part of the integrand.

Change the limits of integration according to the substitution: when \(x = 1\), \(u = \ln 1 = 0\); when \(x = e^2\), \(u = \ln e^2 = 2\).

Rewrite the integral in terms of \(u\): \(\int_0^2 u^5 \, du\), which is a straightforward power integral to evaluate.

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. For example, substituting u = ln(x) can simplify integrals involving logarithmic functions.

Properties of the Natural Logarithm Function

The natural logarithm function, ln(x), is defined for x > 0 and has the derivative 1/x. Understanding its behavior and domain is crucial when integrating expressions involving ln(x), especially when combined with powers or other functions.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two points. When performing substitution, the limits of integration must be adjusted to the new variable, or the integral must be evaluated back in terms of the original variable after integration.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₁ᵉ^² (ln x)^5 / x dx