Evaluate the integral.
from 1 to 5
(ln(x))^2/x^3 dx.
let
u = (lnx)^2
dv = x^-3 dx
du = 2lnx/x dx
v = -1/2 x^-2
∫u dv = uv - ∫v du
∫(lnx)^2/x^3 dx = -(lnx)^2/(2x^2) - ∫(-lnx)/(x^3) dx
Now repeat with u=lnx and you will be left with just a power of x to integrate.
To evaluate the given integral ∫(ln(x))^2/x^3 dx from 1 to 5, we will use the technique of integration by parts. The formula for integration by parts is as follows:
∫u dv = uv - ∫v du
Let's apply this formula to the given integral. We will choose u = ln(x)^2 and dv = 1/x^3 dx.
1. Compute du:
To find du, we differentiate u = ln(x)^2 with respect to x.
du/dx = 2(ln(x))(1/x) = 2ln(x)/x
2. Compute v:
To find v, we integrate dv = 1/x^3 dx.
∫dv = ∫(1/x^3) dx
To integrate 1/x^3 dx, we can rewrite it as x^(-3) and apply the power rule for integration:
∫(1/x^3) dx = -x^(-2)/2 = -1/(2x^2)
Now we have our values for u, du, v, and ∫dv. Substituting these values into the integration by parts formula:
∫(ln(x))^2/x^3 dx = uv - ∫v du
∫(ln(x))^2/x^3 dx = ln(x)^2*(-1/(2x^2)) - ∫[-1/(2x^2)](2ln(x)/x) dx
Simplifying:
∫(ln(x))^2/x^3 dx = -ln(x)^2/(2x^2) + ∫ln(x)/(x^3) dx
We can now break down the integral into two parts:
(a) ∫-ln(x)^2/(2x^2) dx
(b) ∫ln(x)/(x^3) dx
Next, we will evaluate each of these integrals separately.