integral of x/(sqrt((1+x^2)))

解答

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

1 + x^{2} + C

求解步骤

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

使用换元积分法

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

= 1 + x^{2}

解答补常数

= 1 + x^{2} + C

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