integral of cos(x^4+2)

解答

\int cos (x^{4} + 2) d x

- \frac{1}{8} cos (2) x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4})) - sin (2) \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4})) + C

求解步骤

\int cos (x^{4} + 2) d x

使用三角恒等式改写

= \int cos (x^{4}) cos (2) - sin (x^{4}) sin (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 cos (x^{4}) cos (2) d x - \int sin (x^{4}) sin (2) d x

\int cos (x^{4}) cos (2) d x = - \frac{1}{8} cos (2) x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4}))

\int sin (x^{4}) sin (2) d x = sin (2) \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4}))

= - \frac{1}{8} cos (2) x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4})) - sin (2) \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4}))

解答补常数

= - \frac{1}{8} cos (2) x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4})) - sin (2) \frac{1}{8} i x (E_{\frac{3}{4}} (- i x^{4}) - E_{\frac{3}{4}} (i x^{4})) + C

\frac{\partial}{\partial x} (- sin (x^{2} y) \cdot 2 x y)

\int \frac{x}{1 + 4 x ^{2}} d x

\frac{d}{d x} (\frac{x ^{2}}{2 x + 3})

y^{^{'}} + 4 x y = 0

\int 4 (2 x + 5)^{3} d x