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 y/((1+y^2)^{1/2)}

解答

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

解答

(1 + y^{2})^{\frac{1}{2}} + C

求解步骤

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

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{y}{( 1 + y ^{2} ) ^{\frac{1}{2}}} = (y) (1 + y^{2})^{- \frac{1}{2}}

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

使用换元积分法

= \int \frac{1}{2 u ^{\frac{1}{2}}} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{1}{2}}} d u

使用幂法则

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

u = 1 + y^{2}

代回

= \frac{1}{2} \cdot 2 (1 + y^{2})^{\frac{1}{2}}

化简

\frac{1}{2} \cdot 2 (1 + y^{2})^{\frac{1}{2}} : (1 + y^{2})^{\frac{1}{2}}

= (1 + y^{2})^{\frac{1}{2}}

解答补常数

= (1 + y^{2})^{\frac{1}{2}} + C

作图

流行的例子

integral of (xsin(x^2))

\int (x sin (x^{2})) d x

integral from 1 to 4 of 3/(t^4)

\int_{1}^{4} \frac{3}{t ^{4}} d t

y^{''}-2y^'+y=0,y(0)=3,y^'(0)=0

y^{^{''}} - 2 y^{^{'}} + y = 0, y (0) = 3, y^{^{'}} (0) = 0

integral from 0 to 1 of 2pix\sqrt[3]{x}

\int_{0}^{1} 2 π x 3 x d x

derivative of 8x^52pi

\frac{d}{d x} (8 x^{5} 2 π)