∫(e^-x)/ (4-e^(-2x)) dx
the answer = 1/4 ln|4-e^-2x| + C seems to be wrong... I am not sure why though
it is correct, though you may want to try equivalent expressions, such as
1/4 (ln(2 - e^-x) + ln(2 + e^-x))
maybe with absolute-value signs; some say that
∫1/x dx = ln|x| rather than just lnx
disagree, taking the derivative of 1/4 ln|4-e^-2x| does not yield the original
expression. The original numerator would have had to contain e^(-2x)
I tried the following:
let u = e^-x , then du = -e^-x dx, or dx = du / -u
then ∫(e^-x)/ (4-e^(-2x)) dx ---> ∫ u/(4 - u^2) * du/-u
= ∫ -1/(4 - u^2) du
Using partial fractions:
let -1/(4 - u^2) = A/(2+u) + B/(2-u)
A(2-u) + B(2+u) = -1
let u = 2 ----> 4B = -1 , B = -1/4
let u = -2 ----> 4A = -1 , A = -1/4
then -1/(4 - u^2) = -1/4( 1/(2+u) + 1/(2-u)
1/(4 - u^2) = (1/4)( 1/(2+u) + 1/(2-u)
∫ -1/(4 - u^2) du = (1/4)( 1/(2+u) + 1/(2-u) )
= (1/4)( ln(2+u) - ln(2 - u) ) + C
= (1/4)( ln(2 + e^-x) - ln(2 - e^-x) ) + C
looks like a minor error with a negative sign somewhere in Monica's solution,
I would have given her 9/10 for a similar solution.
thank you, I did make an error on the sign.
however when I enter the answer it says that the domain does not match the correct answer.
To find out if the solution you provided is correct, we can differentiate it and see if we obtain the original integrand:
Take the derivative of 1/4 ln|4 - e^(-2x)| + C with respect to x:
d/dx [1/4 ln|4 - e^(-2x)| + C]
Using the chain rule for differentiation and the fact that the derivative of ln(x) is 1/x, we get:
[1/4] * 1/(4 - e^(-2x)) * -2(-e^(-2x))
This simplifies to:
[1/4] * 2e^(-2x)/(4 - e^(-2x))
Now, let's rewrite the integrand of the original integral in a similar form:
e^(-x)/(4 - e^(-2x))
We can see that these two expressions are not equal, implying that the solution you provided is incorrect.
To find the correct solution, we need to use a different approach. One way is to use substitution.
Let's set u = 4 - e^(-2x).
Differentiating both sides with respect to x, we get:
du/dx = 2e^(-2x).
Rearranging this equation, we have:
dx = (1/2e^(-2x)) du.
Now, substitute the new variables into the integral:
∫(e^-x)/(4 - e^(-2x)) dx
= ∫(e^-x)/(u) * (1/2e^(-2x)) du
= (1/2) ∫(1/u) du
Integrating this expression with respect to u, we obtain:
(1/2) ln|u| + C
Now substitute back u = 4 - e^(-2x):
(1/2) ln|4 - e^(-2x)| + C
Hence, the correct solution to the integral is (1/2) ln|4 - e^(-2x)| + C, not 1/4 ln|4 - e^(-2x)| + C as previously stated.