integral of (sin(x)-x)/(x^3)

解答

\int \frac{sin ( x ) - x}{x ^{3}} d x

- \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + \frac{1}{x} + C

求解步骤

\int \frac{sin ( x ) - x}{x ^{3}} d x

乘开

\frac{sin ( x ) - x}{x ^{3}} : \frac{sin ( x )}{x ^{3}} - \frac{1}{x ^{2}}

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

\int \frac{sin ( x )}{x ^{3}} d x = - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x))

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

= - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) - (- \frac{1}{x})

化简

= - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + \frac{1}{x}

解答补常数

= - \frac{sin ( x )}{2 x ^{2}} + \frac{1}{2} (- \frac{cos ( x )}{x} - Si (x)) + \frac{1}{x} + C

y^{^{'}} = 4 (y^{2} + 1)

x \to \infty lim (\frac{1}{x} - 5)

x \to 3 lim (\frac{4}{x ^{2} + 2 x + 3})

\frac{d}{d x} (14 - \frac{1}{2} x^{- 3})

\frac{d}{d x} (cos^{2} (2 x^{4} - 4))