integral from 2 to 6 of sin(x^2-5)

解答

\int_{2}^{6} sin (x^{2} - 5) d x

- sin (5) (\frac{π}{2} C (\frac{2}{π} \cdot 6) - \frac{π}{2} C (\frac{2}{π} \cdot 2)) + cos (5) (\frac{π}{2} § (\frac{2}{π} \cdot 6) - \frac{π}{2} § (\frac{2}{π} \cdot 2))

求解步骤

\int_{2}^{6} sin (x^{2} - 5) d x

使用三角恒等式改写

= \int_{2}^{6} - cos (x^{2}) sin (5) + cos (5) sin (x^{2}) 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

= - \int_{2}^{6} cos (x^{2}) sin (5) d x + \int_{2}^{6} cos (5) sin (x^{2}) d x

\int_{2}^{6} cos (x^{2}) sin (5) d x = sin (5) (\frac{π}{2} C (\frac{2}{π} \cdot 6) - \frac{π}{2} C (\frac{2}{π} \cdot 2))

\int_{2}^{6} cos (5) sin (x^{2}) d x = cos (5) (\frac{π}{2} § (\frac{2}{π} \cdot 6) - \frac{π}{2} § (\frac{2}{π} \cdot 2))

= - sin (5) (\frac{π}{2} C (\frac{2}{π} \cdot 6) - \frac{π}{2} C (\frac{2}{π} \cdot 2)) + cos (5) (\frac{π}{2} § (\frac{2}{π} \cdot 6) - \frac{π}{2} § (\frac{2}{π} \cdot 2))

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