Evaluation and Bound Shifting of Trigonometric Integrals
Consider the definite integral \(I = \int_{0}^{\pi} \sin(x)\,dx\). (a) Evaluate the integral \(I\). (b) Suppose the variable within the sine function is shifted such that \(x = u + \frac{\pi}{4}\). Adjust the bounds of integration accordingly and express the new definite integral.