分部積分法例題編練

Preparing發表於2024-05-25

paradigm 0

\[\begin{align} \int x\cos xdx=? \\ \\ \int x(\sin x)^{\prime}dx \\ \\ 設:u=x, \quad q=\sin x \\ \\ \Rightarrow\int x\left(dx\cdot\sin x\right)= \int x d\left(\sin x\right) \\ \\ =x\sin x-\int\sin xdx \\ \\ =x\sin x-(-\cos x)+C \\ \\ =x\sin x+\cos x+C \end{align} \]


paradigm 1

\[\begin{align} \int xe^xdx=? \\ \\ \because (e^{x})^{\prime}=e^{x}\ln e=e^{x} \\ \\ \Rightarrow\int xd(e^{x})=xe^{x}-\int e^{x}dx \\ \\ =xe^{x}-(e^{x}+C) \\ \\ =e^{x}\left(x-1\right)+C \end{align} \]


paradigm 2

\[\int x^{2} \ln x dx=? \]


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