Quod Erat Demonstrandum

2015/05/14

某積分

Filed under: NSS — johnmayhk @ 5:37 下午
Tags: ,

網上見某中學的積分題,求

\displaystyle \int\frac{e^x(x\ln x+1)}{x}dx

其實出題的 idea 不難,不過是運用

\frac{d}{dx}e^xf(x)=e^x(f(x)+f'(x))

而已。

比如,設

f(x)=\ln x,則 e^x(f(x)+f'(x))=e^x(\ln x+\frac{1}{x}),於是

\int\frac{e^x(x\ln x+1)}{x}dx=e^x\ln x+c(for x > 0

只要玩不同的 f(x),甚至改變一下 e^x 的冪,便可出不是題目,隨便舉例:

1.\displaystyle \int e^x(\sin^{-1}x+\frac{1}{\sqrt{1-x^2}})dx = ?

2.\displaystyle \int \frac{\ln x-x^{-1}}{e^x}dx = ?

3.\displaystyle \int e^{\sin x}(-\sin x+\cos^2 x)dx = ?

諸如此類。

Read also
https://johnmayhk.wordpress.com/2014/08/08/integration-by-comparing-coefficients/

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