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6) ∫[e,e^2] lnx d(-1/(x-1))
= (-1/(x-1)) lnx + ∫[e,e^2] 1/(x(x-1)) dx
= (-1/(x-1)) lnx - ∫[e,e^2] 1/x - 1/(x-1) dx
= (-1/(x-1)) lnx - ln|x/(x-1)| | [e,e^2]
= -2/(e^2-1) + 1/(e-1) - 1 + ln(e+1)
8) ∫[0, √ln2] (-1/2) x^2 de^(-x^2)
= (-1/2) [x^2 e^(-x^2) + e^(-x^2)] | [0, √ln2]
= -(1/2)[(1/2)ln2 - 1/2]
= (1/4)(1-ln2)
= (-1/(x-1)) lnx + ∫[e,e^2] 1/(x(x-1)) dx
= (-1/(x-1)) lnx - ∫[e,e^2] 1/x - 1/(x-1) dx
= (-1/(x-1)) lnx - ln|x/(x-1)| | [e,e^2]
= -2/(e^2-1) + 1/(e-1) - 1 + ln(e+1)
8) ∫[0, √ln2] (-1/2) x^2 de^(-x^2)
= (-1/2) [x^2 e^(-x^2) + e^(-x^2)] | [0, √ln2]
= -(1/2)[(1/2)ln2 - 1/2]
= (1/4)(1-ln2)
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