∫ x^2 to 1 (ln^2x/x)dx=
To solve the integral ∫ x² to 1 (ln²x/x)dx, we can use integration by parts. Integration by parts is a technique used to find the integral of a product of two functions.
The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's assign u and dv to the different parts of our equation. In this case, we can choose:
u = ln²x
dv = 1/x dx
To find du and v, we need to differentiate u and integrate dv, respectively.
Differentiating u:
du = (2 ln x) * (1/x) * dx
Integrating dv:
v = ∫ (1/x) dx
= ln|x|
Now, we have u, dv, du, and v, so we can apply the integration by parts formula:
∫ x² to 1 (ln²x/x)dx = uv - ∫ v du
Plugging in the values we found:
∫ x² to 1 (ln²x/x)dx = ln²x * ln|x| - ∫ ln|x| * (2 ln x) * (1/x) * dx
Simplifying and rearranging:
∫ x² to 1 (ln²x/x)dx = ln²x * ln|x| - 2 ∫ (ln²x/x) dx
Now, we have a similar form of the initial integration problem. We can substitute the original integral back into the equation:
∫ x² to 1 (ln²x/x)dx = ln²x * ln|x| - 2 ∫ x² to 1 (ln²x/x) dx
Next, let's simplify the resulting equation:
∫ x² to 1 (ln²x/x) dx = ln²x * ln |x| - 2 ∫ x² to 1 (ln²x/x) dx
Now, move 2 ∫ x² to 1 (ln²x/x) dx to the other side of the equation:
∫ x² to 1 (ln²x/x) dx + 2 ∫ x² to 1 (ln²x/x) dx = ln²x * ln |x|
Combine the integrals:
3 ∫ x² to 1 (ln²x/x) dx = ln²x * ln |x|
Divide both sides by 3:
∫ x² to 1 (ln²x/x) dx = (ln²x * ln |x|) / 3
So, the solution to the given definite integral is (ln²x * ln |x|) / 3.