20) Evaluate the definite integral. On the integral from
e to e^3
∫dx/xl(nx)^(1/2)
If you mean
dx/x (lnx)^(1/2)
then let
u = lnx
du = dx/x
and the integrand becomes
u^(1/2) du
To evaluate the definite integral, we will start by simplifying the integrand.
The integrand is given as ∫dx/(xl(nx)^(1/2)).
Let's break down the expression inside the square root:
(nx)^(1/2)
We can rewrite this expression as (xn)^(1/2). Applying the power rule of exponents, we have:
(nx)^(1/2) = (x^n)^(1/2) = x^(n/2)
Substituting this back into the integrand, we have:
∫dx/(xl(nx)^(1/2)) = ∫dx/(xl(x^(n/2)))
Next, we can simplify the expression in the denominator. Recall that x^a/x^b = x^(a-b).
So, we have:
∫dx/(xl(x^(n/2))) = ∫dx/(x^(1+(n/2)))
Now, let's focus on the limits of integration, which are from e to e^3:
∫[e to e^3] dx/(x^(1+(n/2)))
To evaluate this definite integral, we need to know the value of n. Without the specific value of n, we cannot calculate the integral. Please provide the value of n to proceed further with the evaluation.