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/(sqrt(x^2-4x+8))

解答

\int \frac{1}{x ^{2} - 4 x + 8} d x

解答

ln (\frac{1}{2} x - 2 + x^{2} - 4 x + 8) + C

求解步骤

\int \frac{1}{x ^{2} - 4 x + 8} d x

对

x^{2} - 4 x + 8

配方:

(x - 2)^{2} + 4

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

使用换元积分法

= \int \frac{1}{u ^{2} + 4} d u

使用三角换元法

= \int sec (v) d v

使用积分常见定则:

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

= ln ∣ tan (v) + sec (v) ∣

代回

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

化简

ln tan (arctan (\frac{1}{2} (x - 2))) + sec (arctan (\frac{1}{2} (x - 2))) : ln (\frac{1}{2} x - 2 + x^{2} - 4 x + 8)

= ln (\frac{1}{2} x - 2 + x^{2} - 4 x + 8)

解答补常数

= ln (\frac{1}{2} x - 2 + x^{2} - 4 x + 8) + C

作图

流行的例子

derivative of 2x^{sqrt(x)}

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

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

\int \frac{2 x - 3}{2 x + 1} d x

integral of-4sin(5x)sin(x)

\int - 4 sin (5 x) sin (x) d x

derivative of 1/(1+sin(x))

\frac{d}{d x} (\frac{1}{1 + sin ( x )})

integral of 3/4 t+6

\int \frac{3}{4} t + 6 d t