integral of cos^2(1/2 x)

解答

\int cos^{2} (\frac{1}{2} x) d x

\frac{1}{2} x + \frac{1}{2} sin (x) + C

求解步骤

\int cos^{2} (\frac{1}{2} x) d x

使用三角恒等式改写

= \int 1 - sin^{2} (\frac{1}{2} x) 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 1 d x - \int sin^{2} (\frac{1}{2} x) d x

\int 1 d x = x

\int sin^{2} (\frac{1}{2} x) d x = \frac{1}{2} (x - sin (x))

= x - \frac{1}{2} (x - sin (x))

化简

x - \frac{1}{2} (x - sin (x)) : \frac{1}{2} x + \frac{1}{2} sin (x)

= \frac{1}{2} x + \frac{1}{2} sin (x)

解答补常数

= \frac{1}{2} x + \frac{1}{2} sin (x) + C

\frac{d}{d x} (7 e^{- 5 x})

\frac{d}{d x} (\frac{10}{e ^{x}})

d er i v a t i v e \frac{6 x ^{2} + 6 x + 8}{x}

\int csc^{4} (9 x) d x

x \to - 1 lim (\frac{x ^{2} - 7}{1 - x})