Evaluate the integrals in Exercises 31–78.
61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv

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Recognize that the integral is of the form \( \int_{1}^{3} \frac{(\ln(v+1))^2}{v+1} \, dv \). To simplify, use the substitution \( u = v + 1 \).

Compute the differential: since \( u = v + 1 \), then \( du = dv \). Also, change the limits of integration accordingly: when \( v = 1 \), \( u = 2 \); when \( v = 3 \), \( u = 4 \).

Rewrite the integral in terms of \( u \): \( \int_{2}^{4} \frac{(\ln u)^2}{u} \, du \).

Recognize that this integral can be approached by using the substitution \( t = \ln u \), which implies \( dt = \frac{1}{u} du \). This transforms the integral into \( \int_{t=\ln 2}^{t=\ln 4} t^2 \, dt \).

Integrate \( t^2 \) with respect to \( t \) over the new limits, then substitute back if needed to express the answer in terms of \( v \).

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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 substituting a part of the integrand with a new variable to transform the integral into a more manageable form. For example, setting u = v + 1 can simplify the given integral by reducing the complexity of the integrand.

Properties of Logarithmic Functions

Understanding logarithmic functions, especially natural logarithms (ln), is essential. The function ln(x) is the inverse of the exponential function e^x, and its properties, such as the chain rule in differentiation and integration, help in manipulating integrals involving ln expressions.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two points. Evaluating a definite integral requires applying the Fundamental Theorem of Calculus, which involves finding an antiderivative and then computing the difference of its values at the upper and lower limits.

Evaluate the integrals in Exercises 31–78.61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv