evaluate the indefinite integral
integral sign [8/(x ln(1x))] dx
= ?????+c
To evaluate the indefinite integral ∫(8/(x ln(1+x))) dx, we can make use of the substitution rule. Let's denote u = ln(1+x).
Now, we need to find du/dx in order to replace dx. Taking the derivative of u = ln(1+x) with respect to x, we get:
du/dx = 1/(1+x)
Rearranging this equation, we have dx = du/(1+x).
Substituting dx and u into the original integral, we get:
∫(8/(x ln(1+x))) dx = ∫(8/(u)) (du/(1+x))
Simplifying, we have:
∫(8/(u(1+x))) du
Next, we will decompose the integrand using partial fraction decomposition. The integrand can be written as:
8/[(u)*(1+x)] = A/u + B/(1+x)
To find the values of A and B, we can find a common denominator for the right side. Multiplying both sides by u*(1+x), we obtain:
8 = A(1+x) + Bu
Expanding and rearranging, we get:
8 = A + Ax + Bu
Comparing coefficients, we have:
A = 8
A + B = 0
Solving this system of equations, we find A = 8 and B = -8. Therefore, our original integrand can be written as:
(8/u) - (8/(1+x))
Now, we can rewrite the integral as:
∫(8/u) du - ∫(8/(1+x)) du
Integrating, we have:
8∫(1/u) du - 8∫(1/(1+x)) du
The integral of (1/u) with respect to u is ln|u|, and the integral of (1/(1+x)) with respect to x is ln|1+x|.
Therefore, our indefinite integral becomes:
8ln|u| - 8ln|1+x| + C
Substituting back u = ln(1+x), we have:
8ln|ln(1+x)| - 8ln|1+x| + C
Thus, the solution to the indefinite integral is:
8ln|ln(1+x)| - 8ln|1+x| + C