integral of (x^2+1)sqrt(2x+1)

解答

\int (x^{2} + 1) 2 x + 1 d x

\frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}} + C

求解步骤

\int (x^{2} + 1) 2 x + 1 d x

使用分布积分法

= \frac{1}{3} (x^{2} + 1) (2 x + 1)^{\frac{3}{2}} - \int \frac{2}{3} x (2 x + 1)^{\frac{3}{2}} d x

\int \frac{2}{3} x (2 x + 1)^{\frac{3}{2}} d x = \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}})

= \frac{1}{3} (x^{2} + 1) (2 x + 1)^{\frac{3}{2}} - \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}})

化简

\frac{1}{3} (x^{2} + 1) (2 x + 1)^{\frac{3}{2}} - \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}}) : \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}}

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}}

解答补常数

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}} + C