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受欢迎的微积分 >

integral of ((1-((x^2)/9)))^{1/2}

解答

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

解答

\frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

求解步骤

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

化简

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

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

提出常数:

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

= \frac{1}{3} \cdot \int (9 - x^{2})^{\frac{1}{2}} d x

使用三角换元法

= \frac{1}{3} \cdot \int 9 cos^{2} (u) d u

提出常数:

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

= \frac{1}{3} \cdot 9 \cdot \int cos^{2} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot 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

= \frac{1}{3} \cdot 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

= \frac{1}{3} \cdot 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)

= \frac{1}{3} \cdot 9 \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

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

代回

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

化简

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

= \frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

解答补常数

= \frac{3}{2} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

作图

流行的例子

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\frac{d}{d y} (x^{2} + y^{3})

integral of (x+2)(x-5)

\int (x + 2) (x - 5) d x

derivative of e^{(x^4/(10)})

\frac{d}{d x} (e^{\frac{x ^{4}}{10}})

(dy)/(dx)=x^2+y-1

\frac{d y}{d x} = x^{2} + y - 1

y^{''}+9y=2sin(2x),y(0)=0,y^'(0)=-1

y^{^{''}} + 9 y = 2 sin (2 x), y (0) = 0, y^{^{'}} (0) = - 1