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 xsqrt(100-(x-10)^2)

解答

\int x 100 - (x - 10)^{2} d x

解答

- \frac{1}{3} (- x^{2} + 20 x)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} x - 1))) + C

求解步骤

\int x 100 - (x - 10)^{2} d x

对

100 - (x - 10)^{2}

配方:

- (x - 10)^{2} + 100

= \int x - (x - 10)^{2} + 100 d x

使用换元积分法

= \int (u + 10) - u^{2} + 100 d u

乘开

(u + 10) - u^{2} + 100 : u - u^{2} + 100 + 10 - u^{2} + 100

= \int u - u^{2} + 100 + 10 - u^{2} + 100 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

= \int u - u^{2} + 100 d u + \int 10 - u^{2} + 100 d u

\int u - u^{2} + 100 d u = - \frac{1}{3} (- u^{2} + 100)^{\frac{3}{2}}

\int 10 - u^{2} + 100 d u = 500 (arcsin (\frac{1}{10} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} u)))

= - \frac{1}{3} (- u^{2} + 100)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} u)))

u = x - 10

代回

= - \frac{1}{3} (- (x - 10)^{2} + 100)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} (x - 10)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x - 10))))

化简

- \frac{1}{3} (- (x - 10)^{2} + 100)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} (x - 10)) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} (x - 10)))) : - \frac{1}{3} (- x^{2} + 20 x)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} x - 1)))

= - \frac{1}{3} (- x^{2} + 20 x)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} x - 1)))

解答补常数

= - \frac{1}{3} (- x^{2} + 20 x)^{\frac{3}{2}} + 500 (arcsin (\frac{1}{10} x - 1) + \frac{1}{2} sin (2 arcsin (\frac{1}{10} x - 1))) + C

作图

流行的例子

integral of (2t^2+1)^{1/3}t^3

\int (2 t^{2} + 1)^{\frac{1}{3}} t^{3} d t

integral of (x^4)/((9+x^5)^2)

\int \frac{x ^{4}}{( 9 + x ^{5} ) ^{2}} d x

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

\int \frac{1}{2} (e^{2 x} - e^{- 2 x}) d x

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

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

integral of 36(x^2+1)(x^3+3x)^2

\int 36 (x^{2} + 1) (x^{3} + 3 x)^{2} d x