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Find the indefinite integral using the substitution provided.


\int\limits({(14e^2^x)/(e^2^x+10)) } \, dx

u=e^2^x+10

User Atahan
by
3.7k points

1 Answer

12 votes

Answer:
7\text{Ln}\left(e^(2x)+10\right)+C

This is the same as writing 7*Ln( e^(2x) + 10) + C

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Step-by-step explanation:

Start with the equation
u = e^(2x)+10

Apply the derivative and multiply both sides by 7 like so


u = e^(2x)+10\\\\(du)/(dx) = 2e^(2x)\\\\7(du)/(dx) = 7*2e^(2x)\\\\7(du)/(dx) = 14e^(2x)\\\\7du = 14e^(2x)dx\\\\

The "multiply both sides by 7" operation was done to turn the 2e^(2x) into 14e^(2x)

This way we can do the following substitutions:


\displaystyle \int (14e^(2x))/(e^(2x)+10)dx\\\\\\\displaystyle \int (1)/(e^(2x)+10)14e^(2x)dx\\\\\\\displaystyle \int (1)/(u)7du\\\\\\\displaystyle 7\int (1)/(u)du\\\\\\

Integrating leads to


\displaystyle 7\int (1)/(u)du\\\\\\7\text{Ln}\left(u\right)+C\\\\\\7\text{Ln}\left(e^(2x)+10\right)+C\\\\\\

Be sure to replace 'u' with e^(2x)+10 since it's likely your teacher wants a function in terms of x. Also, do not forget to have the plus C at the end. This is a common mistake many students forget to do.

To verify the answer, you can apply the derivative to it and you should get back to the original integrand of
(14e^(2x))/(e^(2x)+10)

User George Artemiou
by
3.3k points