Question

∫ sin(2x + 3)dx

Explanation:

Step1: Substitution

Let \( u = 2x + 3 \), then \( du=2dx \), and \( dx=\frac{1}{2}du \).

Step2: Integrate

The integral becomes \(\frac{1}{2}\int\sin(u)du\).
Using \(\int\sin(u)du=-\cos(u)+C\), we get \(\frac{1}{2}(-\cos(u))+C\).

Step3: Back - substitution

Substitute \( u = 2x + 3 \) back, the result is \(-\frac{1}{2}\cos(2x + 3)+C\).

Answer:

\(-\frac{1}{2}\cos(2x + 3)+C\)