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 (3x^2)/(sqrt(49-x^2))

解答

\int \frac{3 x ^{2}}{49 - x ^{2}} d x

解答

\frac{147}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x))) + C

求解步骤

\int \frac{3 x ^{2}}{49 - x ^{2}} d x

提出常数:

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

= 3 \cdot \int \frac{x ^{2}}{49 - x ^{2}} d x

使用三角换元法

= 3 \cdot \int 49 sin^{2} (u) d u

提出常数:

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

= 3 \cdot 49 \cdot \int sin^{2} (u) d u

使用三角恒等式改写

= 3 \cdot 49 \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

= 3 \cdot 49 \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

= 3 \cdot 49 \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)

= 3 \cdot 49 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

u = arcsin (\frac{1}{7} x)

代回

= 3 \cdot 49 \cdot \frac{1}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x)))

化简

3 \cdot 49 \cdot \frac{1}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x))) : \frac{147}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x)))

= \frac{147}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x)))

解答补常数

= \frac{147}{2} (arcsin (\frac{1}{7} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{7} x))) + C

作图

流行的例子

integral of (e^y)/((e^y+1)^2)

\int \frac{e ^{y}}{( e ^{y} + 1 ) ^{2}} d y

y^{^{'}} = \frac{y \cdot e ^{x}}{x}

integral from 3 to 4 of x/(x^2+4x+20)

\int_{3}^{4} \frac{x}{x ^{2} + 4 x + 20} d x

derivative of (4x^2-2x^4)

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

limit as t approaches 0 of (e^{6t}-1)/(sin(t))

t \to 0 lim (\frac{e ^{6 t} - 1}{sin ( t )})