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 5/(9-x^2)

解答

\int \frac{5}{9 - x ^{2}} d x

解答

\frac{5}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

求解步骤

\int \frac{5}{9 - x ^{2}} d x

提出常数:

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

= 5 \cdot \int \frac{1}{9 - x ^{2}} d x

使用换元积分法

= 5 \cdot \int \frac{1}{3 ( - u ^{2} + 1 )} d u

提出常数:

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

= 5 \cdot \frac{1}{3} \cdot \int \frac{1}{- u ^{2} + 1} d u

使用积分常见定则:

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

= 5 \cdot \frac{1}{3} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

u = \frac{x}{3}

代回

= 5 \cdot \frac{1}{3} (\frac{ln \frac{x}{3} + 1}{2} - \frac{ln \frac{x}{3} - 1}{2})

化简

5 \cdot \frac{1}{3} (\frac{ln \frac{x}{3} + 1}{2} - \frac{ln \frac{x}{3} - 1}{2}) : \frac{5}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

= \frac{5}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1)

解答补常数

= \frac{5}{6} (ln \frac{x}{3} + 1 - ln \frac{x}{3} - 1) + C

作图

流行的例子

y^{''}+25y=cos(5x)+sin(5x)

y^{^{''}} + 25 y = cos (5 x) + sin (5 x)

sum from n=1 to infinity of 7/(3^{n-1)}

n = 1 \sum \infty \frac{7}{3 ^{n - 1}}

integral of 1/(x^2+49)

\int \frac{1}{x ^{2} + 49} d x

面积 x^3,3x^2-4

a re a x^{3}, 3 x^{2} - 4

x^3y^'+3x^2y= 1/x ,y(1)=3

x^{3} y^{^{'}} + 3 x^{2} y = \frac{1}{x}, y (1) = 3