Mathematics

Problem

Consider the integral

I=\displaystyle\int (4x-3)\,\sin\bigl[(4x-3)^{2}\bigr] \,dx

. a. (a) Transform

I

into an integral in a single new variable by introducing and later removing an appropriate constant factor. Hence, evaluate

I

. b. (b) Using the antiderivative obtained in part (a), compute

\displaystyle\int_{1}^{2} (4x-3)\,\sin\bigl[(4x-3)^{2}\bigr]\,dx

. Express your final answer as a real number without the constant of integration.