Question

微积分，大神帮帮忙！

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Answer 1

1. G(x) = ∫f(t)dt                  t =0→u(x)

   则 G'(x) = f(x) * u'(x)

                = ln(x²+5) * [e^(2x)]'

                = ln(x²+5) * [2*e^(2x)]

                = 2*ln(x²+5)*e^(2x)

2. 设 u = e^x。则 du = e^x * dx

   那么，上面的不定积分

   =∫(4-e^x)*(e^x * dx)/(e^x + 3)

   =∫(4-u)*du/(u+3)

   =∫[7-(u+3)]*du/(u+3)

   =∫7*du/(u+3) - ∫(u+3)*du/(u+3)

   =7*∫du/(u+3) - ∫du

   =7*∫d(u+3)/(u+3) - u

   =7*ln(u+3) - u + C

   =7*ln(e^x + 3) - e^x + C