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In integration by parts $\displaystyle\int u\,dv$, the preferred order for choosing the first function $u$ is given by the mnemonic ILATE. Which choice does this prescribe for $\int x \sin x\,dx$?
A$u = x,\,dv = \sin x\,dx$
BEither choice gives the same result
C$u = x \sin x,\,dv = dx$
D$u = \sin x,\,dv = x\,dx$
Answer & Solution
Correct answer: A. $u = x,\,dv = \sin x\,dx$
ILATE = Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. The function appearing earlier in this list is chosen as $u$. Between $x$ (algebraic) and $\sin x$ (trigonometric), algebraic comes first, so $u = x$ and $dv = \sin x\,dx$. Choosing the wrong way leaves an integral that is harder than the original.
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