A calculus nightmare

Every IB Math student knows that Calculus plays a big part in the IB Math syllabus. What most people don't know is that AA HL students get a pretty deep dive into the subject, and that could easily replace the first semester of Calculus on a top tier university, as university students will also tackle derivatives and integrals, integration techniques, areas, volumes and differential equations. On that note, we combined areas, volumes and integration by parts into the same question, as an ode to that. Have fun solving this gem!