alleryThe integration-by-parts formula $\int u\,\dfrac{dv}{dx}\,dx$ equals
A$uv+\displaystyle\int v\,\dfrac{du}{dx}\,dx$
B$uv-\displaystyle\int v\,\dfrac{du}{dx}\,dx$
C$\displaystyle\int v\,\dfrac{du}{dx}\,dx-uv$
D$uv-\displaystyle\int u\,\dfrac{du}{dx}\,dx$
Answer & Solution
Correct answer: B. $uv-\displaystyle\int v\,\dfrac{du}{dx}\,dx$
1. From the product rule $\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$.
2. Integrating both sides: $uv=\int u\dfrac{dv}{dx}\,dx+\int v\dfrac{du}{dx}\,dx$.
3. Rearranging gives $\int u\dfrac{dv}{dx}\,dx=uv-\int v\dfrac{du}{dx}\,dx$.
_Source: syllabus Mathematics Ch 7 "Integrals", p.323_
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