integrate by parts
�çx (lnx)^2 dx
integrate by parts
x(lnx)^2
∫u dv = uv - ∫v du
So in ∫x(lnx)^2 dx
let u = (lnx)^2 and dv = x dx
du/dx = 2lnx (1/x) and v = (1/2)x^2
du = (2/x)(lnx) dx and v = (1/2)x^2
∫x(lnx)^2 dx = (1/2)(x^2)(lnx)^2 - ∫(1/2)x^2 (2/x)(lnx) dx
= (1/2)(x^2)(lnx)^2 - ∫(x)(lnx) dx
now let's do ∫(x)(lnx) dx
let u = lnx and dv = x dx
du/dx = 1/x and v = (1/2)x^2
du = (1/x) dx and v = (1/2)x^2
∫(x)(lnx) dx = (lnx)(1/2)(x^2) - ∫(1/2)(x^2)(1/x) dx
= (1/2)(x^2)(lnx) - ∫(1/2)x dx
= (1/2)(x^2)(lnx) - (1/4)x^2
So finally ...
∫x(lnx)^2 dx
= (1/2)(x^2)(lnx)^2 - ∫(x)(lnx) dx
= (1/2)(x^2)(lnx)^2 - ((1/2)(x^2)(lnx) - (1/4)x^2) )
= (1/2)(x^2)(lnx)^2 - (1/2)(x^2)(lnx) + (1/4)x^2)
better check my algebra, I should have written it out on paper first.
To integrate by parts, you will need to employ the product rule of differentiation. The formula for integration by parts is:
∫(u dv) = uv - ∫(v du)
Let's solve the given integral ∫x(lnx)^2 dx using integration by parts.
Step 1: Choose u and dv
In this case, u should be a function that becomes simpler when differentiated, and dv should be a function that can be easily integrated. Let's choose:
u = ln^2x
dv = x dx
Step 2: Find du and v
Differentiate u to find du and integrate dv to find v.
Differentiating u:
du = 2lnx * (1/x) dx
= 2lnx/x dx
Integrating dv:
v = ∫x dx
= (x^2)/2
Step 3: Apply the formula
Using the integration by parts formula:
∫(u dv) = uv - ∫(v du)
∫x(lnx)^2 dx = (ln^2x * (x^2)/2) - ∫((x^2)/2 * 2lnx/x) dx
Simplifying further, we have:
= (x^2(ln^2x))/2 - ∫x(lnx) dx
Step 4: Solve the remaining integral
We have a new integral to solve: ∫x(lnx) dx. In this case, we can use integration by parts again.
Let's choose:
u = ln x (as it simplifies when differentiated)
dv = x dx (as it can be easily integrated)
Differentiating u:
du = (1/x) dx
Integrating dv:
v = ∫x dx
= (x^2)/2
Using the integration by parts formula again:
∫x(lnx) dx = (ln x * (x^2)/2) - ∫((x^2)/2 * (1/x) dx)
Simplifying further, we have:
= (x^2 ln x)/2 - ∫(x/2) dx
= (x^2 ln x)/2 - (x^2)/4 + C
Thus, the final solution to the given integral ∫x(lnx)^2 dx is:
(x^2(ln^2 x))/2 - (x^2 ln x)/2 + (x^2)/4 + C
where C is the constant of integration.