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

解答

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

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

求解步骤

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

提出常数:

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

= 2 \cdot \int x 1 + x^{2} d x

使用换元积分法

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

使用幂法则

= 2 \cdot \frac{u ^{3}}{3}

u = 1 + x^{2}

代回

= 2 \cdot \frac{( 1 + x ^{2} ) ^{3}}{3}

化简

2 \cdot \frac{( 1 + x ^{2} ) ^{3}}{3} : \frac{2}{3} (1 + x^{2})^{\frac{3}{2}}

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

解答补常数

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

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