integral from-2 to 9 of-x*cos(x)

解答

\int_{- 2}^{9} - x \cdot cos (x) d x

- 9 sin (9) - cos (9) + 2 sin (2) + cos (2)

十进制

- 1.39548 \dots

求解步骤

\int_{- 2}^{9} - x cos (x) d x

提出常数:

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

= - \int_{- 2}^{9} x cos (x) d x

使用分布积分法

= - [x sin (x) - \int sin (x) d x]_{- 2}^{9}

\int sin (x) d x = - cos (x)

= - [x sin (x) - (- cos (x))]_{- 2}^{9}

化简

= - [x sin (x) + cos (x)]_{- 2}^{9}

带入上下限计算后得:

9 sin (9) + cos (9) - 2 sin (2) - cos (2)

= - (9 sin (9) + cos (9) - 2 sin (2) - cos (2))

化简

= - 9 sin (9) - cos (9) + 2 sin (2) + cos (2)

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