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受欢迎的微积分 >

integral of x/((1+x^2)^{1/2)}

解答

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

解答

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

求解步骤

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

使用指数法则:

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

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

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

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 + x^{2}

代回

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

化简

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

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

解答补常数

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

作图

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