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(30-10x+x^2)

解答

\int 30 - 10 x + x^{2} d x

解答

5 (\frac{1}{10} (x - 5) x^{2} - 10 x + 30 + \frac{1}{2} ln \frac{1}{5} (x - 5) + \frac{x ^{2} - 10 x + 30}{5}) + C

求解步骤

\int 30 - 10 x + x^{2} d x

对

30 - 10 x + x^{2}

配方:

(x - 5)^{2} + 5

= \int (x - 5)^{2} + 5 d x

使用换元积分法

= \int u^{2} + 5 d u

使用三角换元法

= \int 5 sec^{3} (v) d v

提出常数:

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

= 5 \cdot \int sec^{3} (v) d v

使用降次积分法

= 5 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= 5 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 5 \frac{sec ^{2} ( arctan ( \frac{1}{5} ( x - 5 ) ) ) sin ( arctan ( \frac{1}{5} ( x - 5 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5} (x - 5))) + sec (arctan (\frac{1}{5} (x - 5)))

化简

5 \frac{sec ^{2} ( arctan ( \frac{1}{5} ( x - 5 ) ) ) sin ( arctan ( \frac{1}{5} ( x - 5 ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5} (x - 5))) + sec (arctan (\frac{1}{5} (x - 5))) : 5 (\frac{1}{10} (x - 5) x^{2} - 10 x + 30 + \frac{1}{2} ln \frac{1}{5} (x - 5) + \frac{x ^{2} - 10 x + 30}{5})

= 5 (\frac{1}{10} (x - 5) x^{2} - 10 x + 30 + \frac{1}{2} ln \frac{1}{5} (x - 5) + \frac{x ^{2} - 10 x + 30}{5})

解答补常数

= 5 (\frac{1}{10} (x - 5) x^{2} - 10 x + 30 + \frac{1}{2} ln \frac{1}{5} (x - 5) + \frac{x ^{2} - 10 x + 30}{5}) + C

作图

流行的例子

integral of (2x+5)/(x(x-2))

integral of (x^2)/(sqrt(x^2+a^2))

integral of (1-x^2)^{(-3)/2}

integral of (6x)/((5-3x^2)^2)

integral of (x^2+4x)e^{2x}