Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.∫₁^∞ (lny) / y³ dy

Accepted Answer

Identify the integral to evaluate: $$\$$int_1$$^{\infty} \frac{\ln y}{y^3$} \, dy. $$Notice that the integral has an infinite upper limit, so this is an improper integral and we will need to check for convergence after finding the antiderivative. Consider using integration by parts since the integrand is a product of $$\ln y$$ and $$y^{-3}. $$Let $$u = \ln y ($$which simplifies upon differentiation) and $$dv = y$$^{-3}$$ dy ($$which is straightforward to integrate). Compute the derivatives and antiderivatives for integration by parts: $$du = \frac{1}{y} dy$$ and $$v = \int y$$^{-3}$$ dy = \int y$$^{-3}$$ dy = -\frac{1}{2} y$$^{-2}$$. Apply the integration by parts formula: $$\int u \, dv = uv - \int v \, du$$. Substitute the expressions for u$, v$, du$ to write the integral in terms of simpler integrals. Evaluate the resulting integral and then apply the limits from 1 to $$\infty$$. Since the upper limit is infinite, express the integral as a limit and analyze the behavior of the antiderivative as y$$ 	o \infty$$ to determine convergence.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₁^∞ (lny) / y³ dy

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Key Concepts

Improper Integrals

Integration by Parts

Logarithmic Functions in Integration