I have a calculus homework question centred on integrals and I’d like a clear, step-by-step solution that shows the underlying reasoning, not just the final answer. Please walk me through each manipulation, state any theorems or substitution rules you apply, and typeset the work neatly—LaTeX is fine, a well-scanned handwritten derivation is also acceptable as long as every step is legible. At the end, include a concise check (for example by differentiating the result or using a numerical verification in Wolfram Alpha, MATLAB, or Python-SymPy) so I can be certain the integral was evaluated correctly.