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{11}{35} (2 x + 1)^{\frac{3}{2}} + C

求解步骤

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

乘开

(x^{2} - 1) 2 x + 1 : x^{2} 2 x + 1 - 2 x + 1

= \int x^{2} 2 x + 1 - 2 x + 1 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} 2 x + 1 d x - \int 2 x + 1 d x

\int x^{2} 2 x + 1 d x = \frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}}

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

= \frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} - \frac{1}{3} (2 x + 1)^{\frac{3}{2}}

化简

\frac{1}{105} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} - \frac{1}{3} (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{11}{35} (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{11}{35} (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{11}{35} (2 x + 1)^{\frac{3}{2}} + C

\int cos^{6} (2 x) d x

(y + 1) y^{^{'}} = x y - y

\frac{d y}{d x} = e^{2 y + 8 x}

\frac{d}{d x} (sin^{3} (x^{2} + 7))

in v erse l a pl a ce \frac{1}{2 s ^{2} + 8 s + 10}