Mathematics

Problem

Evaluate the following integrals involving a combination of algebraic and trigonometric functions. a. (i) Compute

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

. b. (ii) Using the antiderivative obtained in part (i), determine the exact value of

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

. Express your answer without any remaining integrals.