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

解

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

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

解答ステップ

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

三角関数の公式を使用して書き換える

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

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

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

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

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