integral from pi/2 to sqrt(x of)sin(15x)

解答

\int_{\frac{π}{2}}^{x} sin (15 x) d x

- \frac{1}{15} cos (15 x)

求解步骤

\int_{\frac{π}{2}}^{x} sin (15 x) d x

使用换元积分法

= \int_{\frac{15 π}{2}}^{15 x} sin (u) \frac{1}{15} d u

提出常数:

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

= \frac{1}{15} \cdot \int_{\frac{15 π}{2}}^{15 x} sin (u) d u

使用积分常见定则:

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

= \frac{1}{15} [- cos (u)]_{\frac{15 π}{2}}^{15 x}

带入上下限计算后得:

- cos (15 x)

= \frac{1}{15} (- cos (15 x))

化简

= - \frac{1}{15} cos (15 x)

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