Integrate
1/(ln(x)); f(e^e)=1
please provide step! confused because of f(e^e)!
To integrate the function 1/(ln(x)), we can use a technique called substitution. Let's start by considering the integral:
∫ (1 / ln(x)) dx
To simplify the expression, we can let u = ln(x). Therefore, du = (1/x) dx.
Substituting these values into the integral, we have:
∫ (1 / ln(x)) dx = ∫ du = u + C,
where C is the constant of integration.
Now, we need to express the answer in terms of x, rather than u. Recall that we initially let u = ln(x).
So, u = ln(x) => e^u = x.
Using this relationship, we can rewrite the integral as:
∫ (1 / ln(x)) dx = u + C = ln(x) + C = ln(e^u) + C = ln(x) + C.
Finally, we need to find the value of C by using the given information that f(e^e) = 1.
Plugging e^e into the expression ln(x) + C, we have:
ln(e^e) + C = e + C = 1.
This equation implies that C = 1 - e.
Therefore, the final solution to the integral is:
∫ (1 / ln(x)) dx = ln(x) + C = ln(x) + (1 - e).