Question

Utiliza el método de integración por partes de la siguiente integral:
$integral \: {x}^{3} \: lnxdx$
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Answer 1

$\int x^3ln\,x\ dx\\\\u=ln\,x\qquad du=\frac{1}{x}dx\\dv=x^3dx \quad v=\frac{1}{4}x^4\\\\\int u\,dv=uv-\int v\,du\\\\\int x^3ln\,x\ dx=(\frac{1}{4}x^4)(ln\,x)-\int \frac{1}{4}x^4\frac{1}{x}\,dx\\\\=\frac{1}{4}x^4\,ln\,x-\frac{1}{4}\int x^3\,dx\\\\=\frac{1}{4}x^4\,ln\,x-\frac{1}{16}x^4\\\\=\frac{x^4}{16}(4\,ln\,x\ -1)$