integral of 1 to e^4 dx/x(1+lnx)

1+lnx = a => dx/x = da
dx/ x(1+lnx) = da/a
int of da/a = In(a)

=>int of dx/ x(1+lnx) = In(1+In(x)) + C

from 1 to e^4 for int of dx/ x(1+lnx) = In(1+In(1)) - In(1+In(e^4))
In(1+In(1)) - In(1+In(e^4)) = -In(5) =~ -1.60944

I am getting this as the wrong answer am i doing something wrong?

It all looks OK to me, except from 1 to e^4 you should have your signs the other way around:

ln(1+ln(e^4)) - ln(1+In(1))

...

=~ +1.60944

Your approach is correct, but there seems to be a mistake in your calculation. Let's go through the steps again:

We want to evaluate the integral:

∫(1 to e^4) dx / x(1+lnx)

Let's start by making the substitution:

1 + lnx = a ⇒ dx / x = da

Therefore, the integral becomes:

∫ da / a

Integrating this gives:

ln|a| + C

Now, we substitute back:

ln|1 + lnx| + C

To evaluate the definite integral from 1 to e^4, we plug in the upper and lower limits:

ln|1 + ln(e^4)| - ln|1 + ln(1)|

Evaluating ln(e^4) simply gives 4, and ln(1) is equal to 0. Our expression simplifies to:

ln(1 + 4) - ln(1) = ln(5) - ln(1) = ln(5)

The final value is ln(5), which is approximately equal to 1.60944. Therefore, the correct answer is approximately 1.60944.

It seems that you made a mistake in your calculation when substituting the upper and lower limits back into the equation. Verify your calculations again, ensuring you accurately compute ln(1) and ln(e^4).