integral from 0 to pi of 19sin^2(4x)

解答

\int_{0}^{π} 19 sin^{2} (4 x) d x

\frac{19 π}{2}

十进制

29.84513 \dots

求解步骤

\int_{0}^{π} 19 sin^{2} (4 x) d x

提出常数:

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

= 19 \cdot \int_{0}^{π} sin^{2} (4 x) d x

使用三角恒等式改写

= 19 \cdot \int_{0}^{π} \frac{1}{2} (1 - cos (8 x)) d x

提出常数:

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

= 19 \cdot \frac{1}{2} \cdot \int_{0}^{π} 1 - cos (8 x) 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

= 19 \cdot \frac{1}{2} (\int_{0}^{π} 1 d x - \int_{0}^{π} cos (8 x) d x)

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (8 x) d x = 0

= 19 \cdot \frac{1}{2} (π - 0)

化简

19 \cdot \frac{1}{2} (π - 0) : \frac{19 π}{2}

= \frac{19 π}{2}

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