积分 (x^2)/(sqrt(x-1))

解答

积分

\frac{x ^{2}}{x - 1}

\frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + 2 x - 1 + C

求解步骤

\int \frac{x ^{2}}{x - 1} d x

使用换元积分法

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

乘开

\frac{( u + 1 ) ^{2}}{u} : u^{\frac{3}{2}} + 2 u + \frac{1}{u}

= \int u^{\frac{3}{2}} + 2 u + \frac{1}{u} d u

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 u^{\frac{3}{2}} d u + \int 2 u d u + \int \frac{1}{u} d u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

\int \frac{1}{u} d u = 2 u

= \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}} + 2 u

u = x - 1

代回

= \frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + 2 x - 1

解答补常数

= \frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + 2 x - 1 + C

\int x^{3} + 3 x^{2} d x

\int csc (x) - sin (x) d x

\int \frac{cos ( 6 x )}{x} d x

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

\int t^{2} e^{9 t} d t