integral from (-x) to x of cos(t^2+1)

解答

\int_{(- x)}^{x} cos (t^{2} + 1) d t

cos (1) (\frac{π}{2} C (\frac{2}{π} x) - \frac{π}{2} C (- \frac{2}{π} x)) - sin (1) (\frac{π}{2} § (\frac{2}{π} x) - \frac{π}{2} § (- \frac{2}{π} x))

求解步骤

\int_{(- x)}^{x} cos (t^{2} + 1) d t

使用三角恒等式改写

= \int_{(- x)}^{x} cos (t^{2}) cos (1) - sin (t^{2}) sin (1) d t

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_{(- x)}^{x} cos (t^{2}) cos (1) d t - \int_{(- x)}^{x} sin (t^{2}) sin (1) d t

\int_{(- x)}^{x} cos (t^{2}) cos (1) d t = cos (1) (\frac{π}{2} C (\frac{2}{π} x) - \frac{π}{2} C (- \frac{2}{π} x))

\int_{(- x)}^{x} sin (t^{2}) sin (1) d t = sin (1) (\frac{π}{2} § (\frac{2}{π} x) - \frac{π}{2} § (- \frac{2}{π} x))

= cos (1) (\frac{π}{2} C (\frac{2}{π} x) - \frac{π}{2} C (- \frac{2}{π} x)) - sin (1) (\frac{π}{2} § (\frac{2}{π} x) - \frac{π}{2} § (- \frac{2}{π} x))

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