Integration of xln(x)

∫ xln(x)dx = ?                 ………………….(1)

Formula to be used: ∫udv = uv – ∫vdu         …………………. (2)

Comparing LHS of equation (1) and (2) i.e. ∫ xln(x)dx with ∫udv, let us assume:

u = ln(x) => du = (1/x) dx …………………. (3)

dv = xdx => v = (x^2)/2 …………………. (4)

Now substituting the values from equation (3) and (4) in the RHS of equation (2) we have:

∫udv = uv – ∫vdu

ð   ∫ xln(x)dx = ((x^2)/2 )ln(x) – ∫((x^2)/2) (1/x) dx

ð  ∫ xln(x)dx = ((x^2)/2 )ln(x) – ∫(x/2) dx

ð  ∫ xln(x)dx = ((x^2)/2 )ln(x) – (x^2)/4 + C

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