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(x^2-10)

解答

\int x^{2} - 10 d x

解答

10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10)) + C

求解步骤

\int x^{2} - 10 d x

使用三角换元法

= \int 10 tan^{2} (u) sec (u) d u

提出常数:

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

= 10 \cdot \int tan^{2} (u) sec (u) d u

使用三角恒等式改写

= 10 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

乘开

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

= 10 \cdot \int - sec (u) + sec^{3} (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

= 10 (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= 10 (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

u = arcsec (\frac{1}{10} x)

代回

= 10 (- ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)) + \frac{1}{2} sec (arcsec (\frac{1}{10} x)) tan (arcsec (\frac{1}{10} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)))

化简

10 (- ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)) + \frac{1}{2} sec (arcsec (\frac{1}{10} x)) tan (arcsec (\frac{1}{10} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x))) : 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10))

= 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10))

解答补常数

= 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10)) + C

作图

流行的例子

limit as x approaches 0-of ((4-x)-0)/4

x \to 0 - lim (\frac{( 4 - x ) - 0}{4})

derivative of 2e^{2x}x

\frac{d}{d x} (2 e^{2 x} x)

integral of e^{3x+1}

\int e^{3 x + 1} d x

integral from 2 to 6 of 1/(x^2-1)

\int_{2}^{6} \frac{1}{x ^{2} - 1} d x

t*(dy)/(dt)=t^3+25t^3y

t \cdot \frac{d y}{d t} = t^{3} + 25 t^{3} y