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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

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∫(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

Source(s): 思瑜的部落格

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