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+2x)

解答

\int x^{2} + 2 x d x

解答

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

求解步骤

\int x^{2} + 2 x d x

对

x^{2} + 2 x

配方:

(x + 1)^{2} - 1

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

使用换元积分法

= \int u^{2} - 1 d u

使用三角换元法

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

使用三角恒等式改写

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

乘开

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

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

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

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

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

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

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

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

(d^2)/(dx^2)(1/(x^7))

\frac{d ^{2}}{d x ^{2}} (\frac{1}{x ^{7}})

limit as x approaches 3 of (x+4)/(x+7)

x \to 3 lim (\frac{x + 4}{x + 7})

integral from 0 to 1 of (x^2+3x+4)

\int_{0}^{1} (x^{2} + 3 x + 4) d x

limit as x approaches-3 of (3-x)/(3+x)

x \to - 3 lim (\frac{3 - x}{3 + x})

derivative 2(ln(x)+1)

d er i v a t i v e 2 (ln (x) + 1)