USE INTEGRATION BY PARTS TO FIND EACH INTEGRAL
ƪ�ãx lnx dx
To find the integral of ƪ�ãx lnx dx using integration by parts, we need to use the formula:
∫ u dv = uv - ∫ v du,
where u and v are functions of x, and du and dv are their respective derivatives.
Let's assign u and dv:
u = lnx (for which du = 1/x dx)
dv = x dx (for which v = (1/2)x^2)
Now, let's substitute these into the formula:
∫ x lnx dx = ∫ u dv
= uv - ∫ v du
= lnx * (1/2)x^2 - ∫ (1/2)x^2 (1/x) dx
= (1/2)x^2 * lnx - (1/2) ∫ x dx
Simplifying further:
= (1/2)x^2 * lnx - (1/2)(1/2)x^2 + C
= (1/2)x^2 * lnx - (1/4)x^2 + C
Therefore, the integral of ƪ�ãx lnx dx is given by (1/2)x^2 * lnx - (1/4)x^2 + C, where C is the constant of integration.