Quick questions on Integration by parts and trigonometric integrals in QCE Specialist Mathematics Unit 4

Integration by parts reverses the product rule. For functions $u$ and $v$,

The order Logarithmic, Inverse-trig, Algebraic, Trigonometric, Exponential (LIATE) is a reliable guide: pick $u$ to be whichever factor appears first in that list, since differentiating it tends to simplify. For $\int x\cos x\, dx$, the algebraic $x$ comes before the trigonometric $\cos x$, so $u = x$ and $dv = \cos x\, dx$.

Sometimes you apply parts twice. For $\int x^2 e^x\, dx$, the first application leaves $\int 2x e^x\, dx$, which needs parts again. Each pass lowers the power of the algebraic factor until it disappears.

A special situation arises with integrands like $\int e^x \sin x\,dx$, where applying parts twice returns a multiple of the original integral. Setting $I = \int e^x\sin x\,dx$, two passes give $I = e^x\sin x - e^x\cos x - I$, and solving algebraically yields $I = \tfrac{1}{2}e^x(\sin x - \cos x) + C$. Recognising the cycle and solving for $I$ rather than integrating endlessly is the key insight, and QCAA expects the algebraic rearrangement to be shown explicitly.

For a definite integral the formula carries the limits on every term: $\displaystyle\int_a^b u\,dv = \big[uv\big]_a^b - \int_a^b v\,du$. Evaluate the boundary term $[uv]_a^b$ at both limits and keep the limits on the remaining integral. A frequent error is to apply the limits only at the end; instead the $uv$ product is evaluated at the limits as soon as it appears, which keeps the bookkeeping clean and avoids losing a boundary contribution.