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 (sqrt(y^2-4))/y

解答

\int \frac{y ^{2} - 4}{y} d y

解答

- 2 arcsec (\frac{1}{2} y) + y^{2} - 4 + C

求解步骤

\int \frac{y ^{2} - 4}{y} d y

使用三角换元法

= \int 2 tan^{2} (u) d u

提出常数:

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

= 2 \cdot \int tan^{2} (u) d u

使用三角恒等式改写

= 2 \cdot \int - 1 + sec^{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 1 d u + \int sec^{2} (u) d u)

\int 1 d u = u

\int sec^{2} (u) d u = tan (u)

= 2 (- u + tan (u))

u = arcsec (\frac{1}{2} y)

代回

= 2 (- arcsec (\frac{1}{2} y) + tan (arcsec (\frac{1}{2} y)))

化简

2 (- arcsec (\frac{1}{2} y) + tan (arcsec (\frac{1}{2} y))) : - 2 arcsec (\frac{1}{2} y) + y^{2} - 4

= - 2 arcsec (\frac{1}{2} y) + y^{2} - 4

解答补常数

= - 2 arcsec (\frac{1}{2} y) + y^{2} - 4 + C

作图

流行的例子

derivative of (x^5-8x^3+x+6/x)

\frac{d}{d x} (\frac{x ^{5} - 8 x ^{3} + x + 6}{x})

limit as x approaches 0 of 2x^3-4x+2

x \to 0 lim (2 x^{3} - 4 x + 2)

integral of (2/x+3sin(x)-4e^x)

\int (\frac{2}{x} + 3 sin (x) - 4 e^{x}) d x

面积 y=sqrt(1+3x),1<= x<= 9

a re a y = 1 + 3 x, 1 \leq x \leq 9

limit as x approaches infinity of a^x

x \to \infty lim (a^{x})