Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 2 of (1 / (y − 1)^(2/3)) dy

Verified step by step guidance

1. Identify the type of integral: This is an improper integral because the integrand \( \frac{1}{(y - 1)^{2/3}} \) becomes undefined at \( y = 1 \), which lies within the interval of integration \([0, 2]\).
2. Split the integral at the point of discontinuity to handle the improper behavior separately: write \( \int_0^2 \frac{1}{(y - 1)^{2/3}} \, dy = \int_0^1 \frac{1}{(y - 1)^{2/3}} \, dy + \int_1^2 \frac{1}{(y - 1)^{2/3}} \, dy \).
3. Rewrite each integral as a limit approaching the point of discontinuity: for the first integral, use \( \lim_{t \to 1^-} \int_0^t \frac{1}{(y - 1)^{2/3}} \, dy \), and for the second integral, use \( \lim_{s \to 1^+} \int_s^2 \frac{1}{(y - 1)^{2/3}} \, dy \).
4. Evaluate the antiderivative of the integrand \( \frac{1}{(y - 1)^{2/3}} \). Use the power rule for integration by rewriting the integrand as \( (y - 1)^{-2/3} \) and then integrate: \( \int (y - 1)^{-2/3} \, dy = \frac{(y - 1)^{1/3}}{(1/3)} + C = 3 (y - 1)^{1/3} + C \).
5. Apply the limits of integration to the antiderivative for each integral and then take the respective limits \( t \to 1^- \) and \( s \to 1^+ \) to determine if the improper integrals converge or diverge.

Key Concepts

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that become unbounded within the interval. To evaluate them, one must consider limits approaching the problematic points to determine if the integral converges or diverges.

Handling Singularities in the Integrand

When the integrand has a singularity (point where it is undefined or infinite) within the integration interval, the integral is split at that point. Each part is evaluated as a limit approaching the singularity to check for convergence.

Integration of Power Functions

Integrating functions of the form (x - a)^p requires applying the power rule for integration, valid when p ≠ -1. For fractional exponents, careful attention is needed to handle the domain and convergence, especially near singularities.