? asked in 科學數學 · 1 decade ago

兩題definite integral超難題

[0,ln2]_∫e ^ (x + e^x) dx

[1,e]_∫x^2 * lnx^2 dx

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  • 1 decade ago
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    ∫(0,㏑2) e^(x+e^x) dx

    = ∫(0,㏑2) e^x * e^e^x dx

    = ∫(0,㏑2) e^e^x d(e^x)

    = e^e^x |(0,㏑2)

    = e^e^㏑2 - e^e^0

    = e² - e¹

    = e² - e

    ---------------------

    ∫(1,e) x²㏑x² dx

    令 u = ㏑x², dv = x² dx

    則 du = 2/x dx, v = x³/3

    ∫ x²㏑x² dx

    = x³㏑x²/3 - ∫ 2x²/3 dx

    = x³㏑x²/3 - 2x³/9 ( 本題為定積分, 故不加入常數 C )

    ∫(1,e) x²㏑x² dx

    = x³㏑x²/3 - 2x³/9 |(1,e)

    = ( e³㏑e²/3 – 2e³/9 ) – ( 1³㏑1²/3 – 2*1³/9 )

    = ( 2e³/3 – 2e³/9 ) – ( 0 – 2/9 )

    = 4e³/9 + 2/9

    http://tw.myblog.yahoo.com/jw!JCVbjQyaBRbXTWOakinc...

    Source(s): 思瑜的部落格
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