An antiderivative of the function is required in a forthcoming analysis

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

Problem

An antiderivative of the function

g(x) = x\,\cos\bigl(x^{2}\bigr)\,e^{\sin\left(x^{2}\right)}

is required in a forthcoming analysis. Let

I = \displaystyle \int x\,\cos\bigl(x^{2}\bigr)\,e^{\sin\left(x^{2}\right)}\,dx.

Answer each part completely, showing all necessary work. (a) Differentiate

h(x)=\sin\bigl(x^{2}\bigr)

and state

h'(x)

. Verify that

h'(x)

can be expressed in the form

k\,x\,\cos\bigl(x^{2}\bigr)

for a constant

k

to be identified. (b) Use your result from part (a) to determine an explicit expression for the indefinite integral

I

. Include the constant of integration. (c) Compute the definite integral

\int_{0}^{\sqrt{\pi/2}} x\,\cos\bigl(x^{2}\bigr)\,e^{\sin\left(x^{2}\right)}\,dx.

Express your answer in exact form. (d) Explain why the integrand is strictly positive on the interval

\left[0,\sqrt{\pi/2}\right]

and interpret the value found in part (c) as the area of a region in the

xy

–plane.