5) Evaluate the definite integral. On the integral from 1 to e^7
∫dx/x(1+lnx)=?
To evaluate the definite integral
∫ dx/x(1 + ln(x))
on the interval from 1 to e^7, we can follow these steps:
Step 1: Simplify the integrand
To simplify the integrand, we can use the properties of logarithms. We have x(1 + ln(x)).
Using the laws of logarithms, we can rewrite this as xln(x) + x.
Step 2: Evaluate the integral
Now, we can rewrite the integral as follows:
∫ dx/x(1 + ln(x)) = ∫ (xln(x) + x) dx/x
Splitting the integral, we can rewrite it as:
∫ xln(x) dx/x + ∫ x dx/x
Step 3: Evaluate each integral separately
∫ xln(x) dx/x = (x^2/2)ln(x) - ∫ (x/2) * (1/x) dx
= (x^2/2)ln(x) - ∫ (1/2) dx
= (x^2/2)ln(x) - x/2 + C1
∫ x dx/x = ∫ dx
= ln|x| + C2
Step 4: Combine the results
Putting the results from Steps 3 into the original equation, we get:
∫ dx/x(1 + ln(x)) = (x^2/2)ln(x) - x/2 + C1 + ln|x| + C2
Simplifying this expression, the final result is:
(x^2/2)ln(x) + ln|x| - x/2 + C
Step 5: Evaluate the definite integral
Finally, we need to evaluate the definite integral on the interval from 1 to e^7. Substituting the limits into the result from Step 4:
[(e^7)^2/2)ln(e^7) + ln|e^7| - e^7/2 + C] - [(1^2/2)ln(1) + ln|1| - 1/2 + C]
Simplifying this expression further, we can calculate the definite integral.