For a real parameter

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

For a real parameter pp, define K(p)=∫2∞x+1xp dx.K(p)=\int_{2}^{\infty} \frac{\sqrt{x}+1}{x^{p}}\,dx. Assume pp is allowed to take any real value. (a) Rewrite the integrand so that K(p)K(p) is expressed as a sum of two simpler improper integrals. State the condition(s) under which each of those integrals is well defined. (b) Determine all real values of pp for which K(p)K(p) converges. Present a clear, logically ordered argument that justifies your conclusion.

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