Integration — FAQ

It's a strategic choice. You want the new integral, $\int v \frac{du}{dx} \, dx$, to be simpler or related to the original integral. For integrals like $\int x^n f(x) \, dx$, choosing $u=x^n$ usually works because differentiating it reduces the power of x. For trigonometric integrals like $\int \sin^n x \, dx$, you split it into $\sin^{n-1} x \cdot \sin x$ and choose $u = \sin^{n-1} x$.

No, this is not a good strategy. Exam questions frequently ask you to 'Show that...' or 'Derive...' the formula itself. Marks are given for the derivation process. You must understand and be able to reproduce the derivation using integration by parts.

No, in this context, the terms 'mean value' and 'average value' of a function over an interval are used interchangeably. Both refer to the value $\frac{1}{b-a} \int_a^b f(x) \, dx$.

There are two likely culprits. First, check your derivatives ($rac{dy}{dx}$ or $rac{dx}{dt}, rac{dy}{dt}$) for simple errors. Second, and more commonly, check if the expression inside the square root simplifies to a perfect square. Exam questions are almost always designed so that this happens, leading to an integral you can solve.