integral from-1 to 1 of pi(x^2-1)^2

解答

\int_{- 1}^{1} π (x^{2} - 1)^{2} d x

π \frac{16}{15}

十进制

3.35103 \dots

求解步骤

\int_{- 1}^{1} π (x^{2} - 1)^{2} d x

提出常数:

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

= π \cdot \int_{- 1}^{1} (x^{2} - 1)^{2} d x

乘开

(x^{2} - 1)^{2} : x^{4} - 2 x^{2} + 1

= π \cdot \int_{- 1}^{1} x^{4} - 2 x^{2} + 1 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= π (\int_{- 1}^{1} x^{4} d x - \int_{- 1}^{1} 2 x^{2} d x + \int_{- 1}^{1} 1 d x)

\int_{- 1}^{1} x^{4} d x = \frac{2}{5}

\int_{- 1}^{1} 2 x^{2} d x = \frac{4}{3}

\int_{- 1}^{1} 1 d x = 2

= π (\frac{2}{5} - \frac{4}{3} + 2)

化简

\frac{2}{5} - \frac{4}{3} + 2 : \frac{16}{15}

= π \frac{16}{15}

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