What is the indefinite integral of xlog(x)?

The integral can be split into two different functions of x which is a hint that we must use the integration by parts method. The method is defined as ∫ uv’ dx = uv - ∫ u’v dx. If we let u = log(x) and v’ = x and then solve for u’ and v such that u’ = 1/x and v = (1/2)x^2 , we can substitute in the values to find the solution. ∫ u’v dx = (1/2)(x^2)log(x) - ∫ (1/x)*((1/2)x^2) dx then goes to ∫ u’v dx = (1/2)(x^2)log(x) - ∫ (1/2)x dx which solves as ∫ u’v dx = (1/2)(x^2)log(x) - (1/4)x^2 + C and can be simplified to read as ∫ u’v dx = (1/4)(x^2)(2log(x) - 1) + C.