integral of sin((pit)/2)

解答

\int sin (\frac{π t}{2}) d t

- \frac{2}{π} cos (\frac{π t}{2}) + C

求解步骤

\int sin (\frac{π t}{2}) d t

使用换元积分法

= \int sin (u) \frac{2}{π} d u

提出常数:

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

= \frac{2}{π} \cdot \int sin (u) d u

使用积分常见定则:

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

= \frac{2}{π} (- cos (u))

u = \frac{π t}{2}

代回

= \frac{2}{π} (- cos (\frac{π t}{2}))

化简

= - \frac{2}{π} cos (\frac{π t}{2})

解答补常数

= - \frac{2}{π} cos (\frac{π t}{2}) + C

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