e to the u Integral Example

[Jason76]

\(\displaystyle \int \dfrac{e^{x}}{(6 - e^{x})^{2}}dx\)

\(\displaystyle \int e^{x} (6 - e^{x})^{-2} dx \)

\(\displaystyle \int e^{x} (u)^{-2} dx \)

\(\displaystyle u = 6 - e^{x}\)

\(\displaystyle du = -e^{x}\)

\(\displaystyle -du = e^{x} dx\)

\(\displaystyle -\int (u)^{-2} dx\)

\(\displaystyle - \dfrac{(u)^{-1}}{-1} dx\)

\(\displaystyle - \dfrac{(6 - e^{x})^{-1}}{-1} + C\)

\(\displaystyle (6 - e^{x})^{-1} + C\) - :?: Not the answer on the computer

 

[Jason76]

Never mind, the computer said it was right. Nonetheless, any interesting observations about this one?

 

[lookagain]

\(\displaystyle \int \dfrac{e^{x}}{(6 - e^{x})^{2}}dx\)

\(\displaystyle \int e^{x} (6 - e^{x})^{-2} dx \)

\(\displaystyle \int e^{x} (u)^{-2} dx \ \ \ \ \)Don't mix u and dx. *

\(\displaystyle u = 6 - e^{x} \ \ \ \ \) Define this *before* substituting it.

\(\displaystyle du = -e^{x} \ \ \ \ \) You're missing the "dx" part.

\(\displaystyle -du = e^{x} dx\)

\(\displaystyle -\int (u)^{-2} dx \ \ \ \ \) See * above.

\(\displaystyle - \dfrac{(u)^{-1}}{-1} dx \ \ \ \ \) There is no dx at this point.

\(\displaystyle - \dfrac{(6 - e^{x})^{-1}}{-1} + C\)

\(\displaystyle (6 - e^{x})^{-1} + C\) - :?: Not the answer on the computer\(\displaystyle \ \ \ \) Did the computer want it expressed with positive exponents?

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