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(9-x^2)))/x

解答

\int \frac{( 9 - x ^{2} )}{x} d x

解答

- \frac{3}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{3}{2} ln \frac{9 - x ^{2}}{3} - 1 + 9 - x^{2} + C

求解步骤

\int \frac{9 - x ^{2}}{x} d x

使用换元积分法

= \int - \frac{u ^{2}}{- u ^{2} + 9} d u

提出常数:

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

= - \int \frac{u ^{2}}{- u ^{2} + 9} d u

使用换元积分法

= - \int \frac{3 v ^{2}}{- v ^{2} + 1} d v

提出常数:

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

= - 3 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 3 \cdot \int \frac{1}{- v ^{2} + 1} - 1 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

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

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

代回

= - 3 \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2} - \frac{9 - x ^{2}}{3}

化简

- 3 \frac{ln \frac{9 - x ^{2}}{3} + 1}{2} - \frac{ln \frac{9 - x ^{2}}{3} - 1}{2} - \frac{9 - x ^{2}}{3} : - \frac{3}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{3}{2} ln \frac{9 - x ^{2}}{3} - 1 + 9 - x^{2}

= - \frac{3}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{3}{2} ln \frac{9 - x ^{2}}{3} - 1 + 9 - x^{2}

解答补常数

= - \frac{3}{2} ln \frac{9 - x ^{2}}{3} + 1 + \frac{3}{2} ln \frac{9 - x ^{2}}{3} - 1 + 9 - x^{2} + C

作图

流行的例子

integral of (8x)/((x-1)^2)

\int \frac{8 x}{( x - 1 ) ^{2}} d x

integral of sec(1/2 x)tan(1/2 x)

\int sec (\frac{1}{2} x) tan (\frac{1}{2} x) d x

integral of ((sqrt(x)+1)^5)/(sqrt(x))

\int \frac{( x + 1 ) ^{5}}{x} d x

integral of (1-\sqrt[3]{x})/(x^2)

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

integral of csc^3(2t-3)

\int csc^{3} (2 t - 3) d t