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

解

\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))

解答ステップ

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

三角関数の公式を使用して書き換える

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

総和規則を適用する:

\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))