integral from 0 to 1/2 of (sin(pix))

解答

\int_{0}^{\frac{1}{2}} (sin (π x)) d x

\frac{1}{π}

十进制

0.31830 \dots

求解步骤

\int_{0}^{\frac{1}{2}} sin (π x) d x

使用换元积分法

= \int_{0}^{\frac{π}{2}} sin (u) \frac{1}{π} d u

提出常数:

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

= \frac{1}{π} \cdot \int_{0}^{\frac{π}{2}} sin (u) d u

使用积分常见定则:

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

= \frac{1}{π} [- cos (u)]_{0}^{\frac{π}{2}}

带入上下限计算后得:

1

= \frac{1}{π} \cdot 1

化简

= \frac{1}{π}

\int_{0}^{2 π} - 36 sin^{2} (t) d t

\int_{0}^{2} 36 t^{2} + 9 t^{4} d t

\int_{0}^{\infty} 1 - e^{- 2 x} d x

\int_{- x}^{x} sin (x) d x

\int_{- \infty}^{0} \frac{2}{x ^{2} + 4} d x