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Calculus/Indefinite Integral

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Expert: - 10/6/2008


Question
What is the indefinite integral of

   1/(1+e^(.2t))

Answer
0.2t is t/5 and will be written as such.

If u=e^(t/5), du = e^(t/5)/5 dt.

Looking at the integral, multiply the top and bottom by e^(t/5)/5, giving (e^(t/5)/5)/((e^(t/5)/5)(1+e^(t/5)).

Making the u substitution gives us du/((u/5)(1+u)), which is the same as 5 du/(u(1+u)).

Let 5/(u(1+u)) = A/u + B/(1+u).

Multiplying through by u(1+u) gives us 5 = A(1+u) + Bu.

From here, we can see that A=5, so B=-5.

This changes the integral to 5/u - 5/(1+u).

The answer is 5ln(u) - 5ln(1+u).

Looking back at what was done, u=e^(t/5), so this converts our problem to t - 5ln(1+e^(t/5)).

Note that in the first one, we had ln(e^x), which is x, but in the second number, it's a bit more complicated by the addition of 1.  And we never thought 1 was that complicated of a number!  

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