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 ((3x-2))/(sqrt(1-x-4x^2))

解答

\int \frac{( 3 x - 2 )}{1 - x - 4 x ^{2}} d x

解答

4 (- \frac{3}{16} - 4 x^{2} - x + 1 - \frac{19}{64} arcsin (\frac{8 x + 1}{17})) + C

求解步骤

\int \frac{3 x - 2}{1 - x - 4 x ^{2}} d x

对

1 - x - 4 x^{2}

配方:

- 4 (x + \frac{1}{8})^{2} + \frac{17}{16}

= \int \frac{3 x - 2}{- 4 ( x + \frac{1}{8} ) ^{2} + \frac{17}{16}} d x

使用换元积分法

= \int \frac{4 ( 3 ( u - \frac{1}{8} ) - 2 )}{- 64 u ^{2} + 17} d u

提出常数:

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

= 4 \cdot \int \frac{3 ( u - \frac{1}{8} ) - 2}{- 64 u ^{2} + 17} d u

乘开

\frac{3 ( u - \frac{1}{8} ) - 2}{- 64 u ^{2} + 17} : \frac{3 u}{- 64 u ^{2} + 17} - \frac{19}{8 - 64 u ^{2} + 17}

= 4 \cdot \int \frac{3 u}{- 64 u ^{2} + 17} - \frac{19}{8 - 64 u ^{2} + 17} 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

= 4 (\int \frac{3 u}{- 64 u ^{2} + 17} d u - \int \frac{19}{8 - 64 u ^{2} + 17} d u)

\int \frac{3 u}{- 64 u ^{2} + 17} d u = - \frac{3}{64} - 64 u^{2} + 17

\int \frac{19}{8 - 64 u ^{2} + 17} d u = \frac{19}{64} arcsin (\frac{8}{17} u)

= 4 (- \frac{3}{64} - 64 u^{2} + 17 - \frac{19}{64} arcsin (\frac{8}{17} u))

u = x + \frac{1}{8}

代回

= 4 - \frac{3}{64} - 64 (x + \frac{1}{8})^{2} + 17 - \frac{19}{64} arcsin (\frac{8}{17} (x + \frac{1}{8}))

化简

4 - \frac{3}{64} - 64 (x + \frac{1}{8})^{2} + 17 - \frac{19}{64} arcsin (\frac{8}{17} (x + \frac{1}{8})) : 4 (- \frac{3}{16} - 4 x^{2} - x + 1 - \frac{19}{64} arcsin (\frac{8 x + 1}{17}))

= 4 (- \frac{3}{16} - 4 x^{2} - x + 1 - \frac{19}{64} arcsin (\frac{8 x + 1}{17}))

解答补常数

= 4 (- \frac{3}{16} - 4 x^{2} - x + 1 - \frac{19}{64} arcsin (\frac{8 x + 1}{17})) + C

作图

流行的例子

integral of piysin(pixy)

\int π y sin (π x y) d x

integral of (e^{6x})

\int (e^{6 x}) d x

integral of-sin(2x)cos(x)

\int - sin (2 x) cos (x) d x

integral of sin^3(x/5)cos(x/5)

\int sin^{3} (\frac{x}{5}) cos (\frac{x}{5}) d x

integral of 6/(x^2+3)

\int \frac{6}{x ^{2} + 3} d x