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(9-x^2))(1)

解答

\int \frac{x ^{2}}{9 - x ^{2}} (1) d x

解答

\frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + C

求解步骤

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

化简

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

使用三角换元法

= \int 9 sin^{2} (u) d u

提出常数:

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

= 9 \cdot \int sin^{2} (u) d u

使用三角恒等式改写

= 9 \cdot \int \frac{1 - cos ( 2 u )}{2} d u

提出常数:

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

= 9 \cdot \frac{1}{2} \cdot \int 1 - cos (2 u) 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

= 9 \cdot \frac{1}{2} (\int 1 d u - \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= 9 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

u = arcsin (\frac{1}{3} x)

代回

= 9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

化简

9 \cdot \frac{1}{2} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) : \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2})

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2})

解答补常数

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + C

作图

流行的例子

(\partial)/(\partial y)((ye^x)^2)

\frac{\partial}{\partial y} ((y e^{x})^{2})

f(x)=(x^{-2}+x)^3(4x)^{-2x}

f (x) = (x^{- 2} + x)^{3} (4 x)^{- 2 x}

integral of sin(8x)ln(sin(8x))

\int sin (8 x) ln (sin (8 x)) d x

integral of (cos(x)-xsin(x))/(xcos(x))

\int \frac{cos ( x ) - x sin ( x )}{x cos ( x )} d x

derivative of ln((x+sqrt(x^2+4)/2))

\frac{d}{d x} (ln (\frac{x + x ^{2} + 4}{2}))