Họ nguyên hàm của hàm số \(f\left(x\right)=x\left(1+x^2\right)^{\frac{3}{2}}\) là
\(\frac{1}{5}\left(1+x^2\right)^{\frac{2}{5}}+C\). \(\frac{2}{5}x\left(1+x^2\right)^{\frac{5}{2}}+C\). \(\frac{1}{5}\left(1+x^2\right)^{\frac{5}{2}}+C\). \(\frac{2}{3}x^2+\frac{1}{4}x^4+C\). Hướng dẫn giải:Đặt \(u=1+x^2\) suy ra \(\text{d}u=2x\text{d}x\) .
\(\int x\left(1+x^2\right)^{\dfrac{3}{2}}\text{d}x=\dfrac{1}{2}\int\left(1+x^2\right)^{\dfrac{3}{2}}\left(2x\text{\text{d}x}\right)=\dfrac{1}{2}\int u^{\dfrac{3}{2}}\text{du}\)
\(=\dfrac{1}{2}.\dfrac{1}{\frac{3}{2}+1}u^{\frac{3}{2}+1}+C=\dfrac{1}{5}u^{\frac{5}{2}}+C\)
\(=\dfrac{1}{5}\left(1+x^2\right)^{\frac{5}{2}}+C\).
