Thursday, March 24, 2022

March Calculus Madness Sweet Sixteen - Day 9: ∫ e^x/(e^x + 1) dx and ∫ e^x/(e^x - 1) dx

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Welcome to March Calculus Madness!


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∫ e^x/(e^x + 1) dx


Let z = e^x

Then: 

ln z = x

1/z dz = dx


∫ e^x/(e^x + 1) dx

= ∫ z/(z + 1) * 1/z dz

= ∫ 1/(z + 1) dz

= ln |z + 1| + C

= ln |e^x + 1| + C


∫ e^x/(e^x - 1) dx


Again, let z = e^x


∫ e^x/(e^x - 1) dx

= ∫ z/(z - 1) * 1/z dz

= ∫ 1/(z - 1) dz

= ln |z - 1| + C

= ln |e^x - 1| + C


Eddie 



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