Question

Tính nguyên hàm \(\int xln\left(x+1\right)dx\)

Answer 1

\(\left\{{}\begin{matrix}u=ln\left(x+1\right)\\dv=xdx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=\dfrac{dx}{x+1}\\v=\dfrac{1}{2}x^2\end{matrix}\right.\)

\(\Rightarrow\int xln\left(x+1\right)dx=\dfrac{1}{2}x^2.ln\left(x+1\right)-\dfrac{1}{2}\int\dfrac{x^2}{x+1}dx\)

\(\int\dfrac{x^2dx}{x+1}=\int\left(x-1\right)dx+\int\dfrac{dx}{x+1}\)

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