5) Evaluate the definite integral. On the integral from 1 to e^7
∫dx/x(1+lnx)
Let u = 1+lnx
du = dx/x
and the integrand is
u du
To evaluate the definite integral ∫ dx/x(1+ln(x)) from 1 to e^7, we can use a technique known as integration by substitution.
Step 1: Let u = 1 + ln(x).
Taking the derivative of both sides, we get du/dx = 1/x.
Rearranging the terms, we can rewrite this as dx = x du.
Step 2: Substitute the values in the integral:
∫ dx/x(1+ln(x)) = ∫ (xdu)/(u).
To change the limits of integration, we substitute the given limits of x = 1 and x = e^7 into u = 1 + ln(x).
When x = 1, u = 1 + ln(1) = 1
When x = e^7, u = 1 + ln(e^7) = 1 + 7 = 8.
So, the new integral becomes:
∫ (xdu)/(u) = ∫ du = u.
Step 3: Evaluate the integral with the new limits:
The integral u evaluated from 1 to 8 is:
∫ 1 du = u | from 1 to 8 = 8 - 1 = 7.
Therefore, the value of the definite integral ∫ dx/x(1+ln(x)) from 1 to e^7 is 7.