∫ x^2 to 1 (ln^2x/x)dx=
(A)(7/3e^2)
b)0
c)2.72
d)4.44
e)1.72
To solve the given integral ∫ x^2 to 1 (ln^2x/x) dx, we need to use integration by parts. Integration by parts is based on the product rule of differentiation.
Let's break down the steps to find the solution:
Step 1: Choose the proper parts
We need to choose two functions: u and dv, to apply the integration by parts formula. In this case, let's choose:
u = ln^2x (which can also be written as (lnx)^2)
dv = 1/x dx
Step 2: Compute the derivatives
To obtain the first derivative of u, we use the chain rule, which states:
d(u)/dx = (d(ln^2x)/dx)
Differentiating (ln^2x) with respect to x, we get:
d(u)/dx = 2lnx * (1/x)
To obtain the antiderivative of dv, we can simply integrate it:
v = ∫(dv) = ∫(1/x dx) = ln|x|
Step 3: Apply the integration by parts formula
The integration by parts formula is given by:
∫udv = uv - ∫vdu
Applying this formula, we have:
∫ x^2 to 1 (ln^2x/x) dx = (ln^2x * ln|x|) - ∫(2lnx * ln|x| / x) dx
Step 4: Simplify and evaluate the integral
Now our goal is to simplify the new integral:
∫(2lnx * ln|x| / x) dx
To simplify further, we can use a property of logarithms which states:
ln(a * b) = ln(a) + ln(b)
Using this property, we can rewrite the integral:
∫(2lnx * ln|x| / x) dx = 2∫(lnx * ln|x| / x) dx
Let's denote the new integral as I:
I = ∫(lnx * ln|x| / x) dx
To solve this integral, we can again use integration by parts.
Choose:
u = lnx
dv = ln|x| / x dx
Compute:
d(u)/dx = 1/x
v = ∫(ln|x| / x dx) = (1/2)(ln^2|x|)
Apply the integration by parts formula:
I = (lnx)(1/2)(ln^2|x|) - ∫((1/2)(ln^2|x|) / x) dx
Simplifying further:
I = (lnx)(1/2)(ln^2|x|) - (1/2)∫(ln^2|x| / x) dx
Notice that the second term in the last equation is the same integral we started with (I). So, substituting (I) in the equation, we get:
I = (lnx)(1/2)(ln^2|x|) - (1/2)I
Now, let's solve for I:
I + (1/2)I = (lnx)(1/2)(ln^2|x|)
(3/2)I = (lnx)(1/2)(ln^2|x|)
2I = 3(lnx)(ln^2|x|)
Finally, integrate I:
∫(lnx * ln|x| / x) dx = (3/2)(lnx)(ln^2|x|) + C
Substituting this result back into the original integral, we get:
∫ x^2 to 1 (ln^2x/x) dx = (ln^2x * ln|x|) - (3/2)(lnx)(ln^2|x|) + C
Now we can evaluate the definite integral by plugging in the limits of integration (x = 1 and x = x^2):
∫ x^2 to 1 (ln^2x/x) dx = [(ln^2(x^2) * ln|x^2|) - (3/2)(ln(x^2))(ln^2|x^2|)] - [(ln^2(1) * ln|1|) - (3/2)(ln(1))(ln^2|1|)]
Simplifying further:
∫ x^2 to 1 (ln^2x/x) dx = [(ln^2(x^2) * ln|x^2|) - (3/2)(ln(x^2))(ln^2|x^2|)] - 0
At this point, we can simplify the expressions inside the brackets and calculate the final result.