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受欢迎的微积分 >

integral from 0 to 1 of (x^2+1)^{10}(2x)

解答

\int_{0}^{1} (x^{2} + 1)^{10} (2 x) d x

解答

\frac{2047}{11}

+1

十进制

186.09090 \dots

求解步骤

\int_{0}^{1} (x^{2} + 1)^{10} \cdot 2 x d x

提出常数:

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

= 2 \cdot \int_{0}^{1} (x^{2} + 1)^{10} x d x

使用换元积分法

= 2 \cdot \int_{1}^{2} \frac{u ^{10}}{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_{1}^{2} u^{10} d u

使用幂法则

= 2 \cdot \frac{1}{2} [\frac{u ^{11}}{11}]_{1}^{2}

化简

2 \cdot \frac{1}{2} [\frac{u ^{11}}{11}]_{1}^{2} : [\frac{u ^{11}}{11}]_{1}^{2}

= [\frac{u ^{11}}{11}]_{1}^{2}

带入上下限计算后得:

\frac{2047}{11}

= \frac{2047}{11}

作图

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