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受欢迎的微积分 >

integral of 1/(sqrt((x-1)(2-x)))

解答

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

解答

arcsin (2 x - 3) + C

求解步骤

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

使用换元积分法

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

乘开

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

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

对

- u^{2} + u

配方:

- (u - \frac{1}{2})^{2} + \frac{1}{4}

= \int \frac{1}{- ( u - \frac{1}{2} ) ^{2} + \frac{1}{4}} d u

使用换元积分法

= \int \frac{2}{- 4 v ^{2} + 1} d v

提出常数:

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

= 2 \cdot \int \frac{1}{- 4 v ^{2} + 1} d v

使用三角换元法

= 2 \cdot \int \frac{1}{2} d w

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int 1 d w

常数积分:

\int a d x = a x

= 2 \cdot \frac{1}{2} \cdot 1 \cdot w

代回

= 2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (2 (x - 1 - \frac{1}{2}))

化简

2 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (2 (x - 1 - \frac{1}{2})) : arcsin (2 x - 3)

= arcsin (2 x - 3)

解答补常数

= arcsin (2 x - 3) + C

作图

流行的例子

integral from-3 to 3 of x^2-3

\int_{- 3}^{3} x^{2} - 3 d x

斜率 y=x^3-3x

s l o p e y = x^{3} - 3 x

derivative of 2*x*y

\frac{d}{d x} (2 \cdot x \cdot y)

integral of (e^x)/2

\int \frac{e ^{x}}{2} d x

d/(dt)(acos(2t)+bsin(2t))

\frac{d}{d t} (a cos (2 t) + b sin (2 t))