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問題
計算
\(\displaystyle\int\log(x^3+x)dx\)
\(x\log(x^3+x)-\displaystyle\int\displaystyle\frac{3x^2+1}{x^3+x}\cdot x dx\) ※部分積分
\(=x\log(x^3+x)-\)\(\displaystyle\int\displaystyle\frac{3x^2+1}{x^2+1}dx\)
\(=x\log(x^3+x)-\)\(\displaystyle\int\displaystyle\frac{3(x^2+1)-2}{x^2+1}dx\)
\(=x\log(x^3+x)-\)\(\displaystyle\int\biggl(3-\displaystyle\frac{2}{x^2+1}\biggr)dx\)
\(=x\log(x^3+x)-3x+2\tan^{-1} x+C\)
答え
\(x\log(x^3+x)-3x+2\tan^{-1} x+C\)
