For a fixed real parameter

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

Problem

a>0

, define

P(a)=\displaystyle \int_{0}^{a^{1/2}} \frac{x\,e^{x^{2}}}{\bigl(1+e^{x^{2}}\bigr)^2}\,dx

. The integral

P(a)

appears in logistic growth models that include an exponential response to a quadratic stimulus. (1) Part 1: (a) Evaluate the indefinite integral

\displaystyle \int \frac{x\,e^{x^{2}}}{\bigl(1+e^{x^{2}}\bigr)^2}\,dx

and express your antiderivative in its simplest exact form. (2) Part 2: (b) Hence obtain a closed–form expression for

P(a)

in terms of

a

. (3) Part 3: (c) Prove that

\displaystyle \lim_{a\to \infty} P(a)=\dfrac12.

(4) Part 4: (d) The function

p(a)=P(a)

can be interpreted as a cumulative distribution function (CDF). Verify that

p(a)

satisfies the defining properties of a CDF and determine its corresponding probability density function (PDF).