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 (x^2)/(sqrt(10x-x^2))

解答

\int \frac{x ^{2}}{10 x - x ^{2}} d x

解答

\frac{25}{4} (2 arcsin (\frac{1}{5} x - 1) - sin (2 arcsin (\frac{1}{5} x - 1))) - 10 - x^{2} + 10 x + 25 arcsin (\frac{1}{5} x - 1) + C

求解步骤

\int \frac{x ^{2}}{10 x - x ^{2}} d x

对

10 x - x^{2}

配方:

- (x - 5)^{2} + 25

= \int \frac{x ^{2}}{- ( x - 5 ) ^{2} + 25} d x

使用换元积分法

= \int \frac{( u + 5 ) ^{2}}{- u ^{2} + 25} d u

乘开

\frac{( u + 5 ) ^{2}}{- u ^{2} + 25} : \frac{u ^{2}}{- u ^{2} + 25} + \frac{10 u}{- u ^{2} + 25} + \frac{25}{- u ^{2} + 25}

= \int \frac{u ^{2}}{- u ^{2} + 25} + \frac{10 u}{- u ^{2} + 25} + \frac{25}{- u ^{2} + 25} 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 \frac{u ^{2}}{- u ^{2} + 25} d u + \int \frac{10 u}{- u ^{2} + 25} d u + \int \frac{25}{- u ^{2} + 25} d u

\int \frac{u ^{2}}{- u ^{2} + 25} d u = \frac{25}{2} (arcsin (\frac{1}{5} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} u)))

\int \frac{10 u}{- u ^{2} + 25} d u = - 10 - u^{2} + 25

\int \frac{25}{- u ^{2} + 25} d u = 25 arcsin (\frac{1}{5} u)

= \frac{25}{2} (arcsin (\frac{1}{5} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} u))) - 10 - u^{2} + 25 + 25 arcsin (\frac{1}{5} u)

u = x - 5

代回

= \frac{25}{2} (arcsin (\frac{1}{5} (x - 5)) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} (x - 5)))) - 10 - (x - 5)^{2} + 25 + 25 arcsin (\frac{1}{5} (x - 5))

化简

\frac{25}{2} (arcsin (\frac{1}{5} (x - 5)) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} (x - 5)))) - 10 - (x - 5)^{2} + 25 + 25 arcsin (\frac{1}{5} (x - 5)) : \frac{25}{4} (2 arcsin (\frac{1}{5} x - 1) - sin (2 arcsin (\frac{1}{5} x - 1))) - 10 - x^{2} + 10 x + 25 arcsin (\frac{1}{5} x - 1)

= \frac{25}{4} (2 arcsin (\frac{1}{5} x - 1) - sin (2 arcsin (\frac{1}{5} x - 1))) - 10 - x^{2} + 10 x + 25 arcsin (\frac{1}{5} x - 1)

解答补常数

= \frac{25}{4} (2 arcsin (\frac{1}{5} x - 1) - sin (2 arcsin (\frac{1}{5} x - 1))) - 10 - x^{2} + 10 x + 25 arcsin (\frac{1}{5} x - 1) + C

作图

流行的例子

integral of 5x^2+sqrt(x)

\int 5 x^{2} + x d x

integral of ((x^3+2x-7)/x)

\int (\frac{x ^{3} + 2 x - 7}{x}) d x

integral of (cos(x)+1/x+pix)

\int (cos (x) + \frac{1}{x} + π x) d x

integral of ((2-x^2))/(x^4)

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

integral of (x+4)/(x^2+4x+6)

\int \frac{x + 4}{x ^{2} + 4 x + 6} d x