Resposta:

Explicação passo a passo:

Calcular a integral indefinida de

    [tex]\displaystyle\int (x\ln x+x\ln^2 x)\,dx\\\\\\=\int x\ln x\cdot (1+\ln x)\,dx\qquad\mathrm{(i)}[/tex]

Faça a seguinte substituição:

    [tex]\begin{array}{ll}x\ln x=u\quad&\Longrightarrow\quad \dfrac{d}{dx}(x\ln x)\,dx=du\\\\ &\Longleftrightarrow\quad \left(\dfrac{d}{dx}(x)\cdot \ln x+x\cdot \dfrac{d}{dx}(\ln x)\right)\!dx=du\\\\\\&\Longleftrightarrow\quad \left(1\cdot \ln x+x\cdot \dfrac{1}{x}\right)\!dx=du\\\\\\ &\Longleftrightarrow\quad (\ln x+1)\,dx=du \end{array}[/tex]

Substituindo, a integral fica

    [tex]\displaystyle= \int u\,du\\\\\\=\dfrac{u^2}{2}+C\\\\\\\\=\dfrac{(x\ln x)^2}{2}+C\quad\longleftarrow\quad\mathsf{resposta.}[/tex]

Dúvidas? Comente.

Bons estudos!