integral of (sqrt(1-\sqrt{x)})

解答

\int (1 - x) d x

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

求解步骤

\int 1 - x d x

使用换元积分法

= \int - 2 (- u + 1) u d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int (- u + 1) u d u

乘开

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

= - 2 \cdot \int - u^{\frac{3}{2}} + 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

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

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

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

= - 2 (- \frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

u = 1 - x

代回

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

解答补常数

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

l a pl a ce t r an s f or m e^{- 4 t} \times sin (2 t)

t an g e n t \frac{10}{3} x^{3} - 5 x^{2} - 11 x + 6

\int \frac{x}{100 x ^{2} - 1} d x

im pl i c i t \frac{d y}{d x}, x = y^{18}

d er i v a t i v e g (x) = 1 - x