zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of 1/((1-x)^3sqrt(x^2-2x-1))

解答

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

解答

- \frac{1}{8 2} (2 arcsec (\frac{1}{2} (x - 1)) + sin (2 arcsec (\frac{1}{2} (x - 1)))) + C

求解步骤

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

分解

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

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

提出常数:

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

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

使用换元积分法

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

使用换元积分法

= - \int \frac{8}{v ^{3} v ^{2} - 8} d v

提出常数:

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

= - 8 \cdot \int \frac{1}{v ^{3} v ^{2} - 8} d v

使用三角换元法

= - 8 \cdot \int \frac{1}{16 2 sec ^{2} ( w )} d w

提出常数:

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

= - 8 \cdot \frac{1}{16 2} \cdot \int \frac{1}{sec ^{2} ( w )} d w

使用三角恒等式改写

= - 8 \cdot \frac{1}{16 2} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

提出常数:

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

= - 8 \cdot \frac{1}{16 2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

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

= - 8 \cdot \frac{1}{16 2} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= - 8 \cdot \frac{1}{16 2} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

代回

= - 8 \cdot \frac{1}{16 2} \cdot \frac{1}{2} (arcsec (\frac{1}{2 2} \cdot 2 (x - 1)) + \frac{1}{2} sin (2 arcsec (\frac{1}{2 2} \cdot 2 (x - 1))))

化简

- 8 \cdot \frac{1}{16 2} \cdot \frac{1}{2} (arcsec (\frac{1}{2 2} \cdot 2 (x - 1)) + \frac{1}{2} sin (2 arcsec (\frac{1}{2 2} \cdot 2 (x - 1)))) : - \frac{1}{8 2} (2 arcsec (\frac{1}{2} (x - 1)) + sin (2 arcsec (\frac{1}{2} (x - 1))))

= - \frac{1}{8 2} (2 arcsec (\frac{1}{2} (x - 1)) + sin (2 arcsec (\frac{1}{2} (x - 1))))

解答补常数

= - \frac{1}{8 2} (2 arcsec (\frac{1}{2} (x - 1)) + sin (2 arcsec (\frac{1}{2} (x - 1)))) + C

作图

流行的例子

derivative \sqrt[3]{x}+2sqrt(x^3)

d er i v a t i v e 3 x + 2 x^{3}

5(dx)/(dt)+10x=120sin(60t)

5 \frac{d x}{d t} + 10 x = 120 sin (60 t)

tangent f(x)= 1/(x-2),\at x=6

t an g e n t f (x) = \frac{1}{x - 2}, at x = 6

integral of 1/(2sin(x)-cos(x)+5)

\int \frac{1}{2 sin ( x ) - cos ( x ) + 5} d x

derivative of (sqrt(5)^x)

\frac{d}{d x} ((5)^{x})