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受欢迎的微积分 >

integral of 2x(x^2+1)^9

解答

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

解答

\frac{1}{10} (x^{2} + 1)^{10} + C

求解步骤

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

提出常数:

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

= 2 \cdot \int x (x^{2} + 1)^{9} d x

使用换元积分法

= 2 \cdot \int \frac{u ^{9}}{2} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{9} d u

使用幂法则

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{10}}{10}

u = x^{2} + 1

代回

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{10}}{10}

化简

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{10}}{10} : \frac{1}{10} (x^{2} + 1)^{10}

= \frac{1}{10} (x^{2} + 1)^{10}

解答补常数

= \frac{1}{10} (x^{2} + 1)^{10} + C

作图

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