Evaluate the definite integral.
dx/(x*sqrt(lnx)) from 1 to e^4
I'm having trouble figuring this out, I might be overthinking it. The way I've approached this is by using u substitution and letting u=lnx. I keep getting the wrong answer though.
Help is appreciated!!
For the integral I go
2 (ln(x))^(1/2) or 2√( ln(x) )
Got it by playing around with it
I will let you do the substitutions.
To evaluate the definite integral of dx/(x*sqrt(lnx)) from 1 to e^4, you can indeed use the method of substitution. Let's review the steps together:
Step 1: Let u = ln(x).
Taking the derivative of both sides, du = (1/x) dx.
Step 2: Rearrange the equation to solve for dx:
dx = x du.
Step 3: Substitute the values of du and dx back into the integral:
∫ (dx/(x*sqrt(lnx))) = ∫ (x du)/(x*sqrt(u)) = ∫ (1/sqrt(u)) du.
Step 4: Simplify the integral:
∫ (1/sqrt(u)) du = 2√u.
Step 5: Substitute the value of u back into the solution:
= 2√(ln(x)).
Step 6: Now, evaluate the integral from 1 to e^4:
= 2√(ln(e^4)) - 2√(ln(1))
= 2√(4) - 2√(0)
= 2(2) - 2(0)
= 4.
Therefore, the definite integral of dx/(x*sqrt(lnx)) from 1 to e^4 is equal to 4.