integral from 1 to e^5

1/(x(sqrt4+lnx))dx

To evaluate the integral ∫(1/(x(sqrt(4+lnx))))dx from 1 to e^5, we can use the substitution method. Here's how to do it step by step:

Step 1: Let u = 4 + ln(x).
Taking the derivative with respect to x, we have du/dx = 1/x.

Step 2: Rearrange the equation to solve for dx:
dx = du/(1/x)
= x du.

Step 3: Substitute the value of dx into the original integral:
∫ (1/(x(sqrt(4+lnx))))dx
= ∫ (1/(x√u)) x du
= ∫ (1/√u) du.

Step 4: Integrate with respect to u:
∫ (1/√u) du
= 2√u + C,
where C is the constant of integration.

Step 5: Substitute the original variable back in terms of u:
2√u + C
= 2√(4 + ln(x)) + C.

Step 6: Evaluate the integral from 1 to e^5:
[2√(4 + ln(x)) + C] evaluated from 1 to e^5
= [2√(4 + ln(e^5)) + C] - [2√(4 + ln(1)) + C]
= 2√(4 + 5) - 2√(4 + 0)
= 2√9 - 2√4
= 6 - 4
= 2.

Therefore, the value of the integral from 1 to e^5 of 1/(x(sqrt(4+lnx)))dx is 2.