What is the integral of (sin(x)-x)/(x^3) ?

The integral of (sin(x)-x)/(x^3) is -(sin(x)}{2x^2}+1/2 (-\frac{cos(x))/x-\Si(x))+1/x+C

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

\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

Solution steps

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

Expand

\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

Apply the Sum Rule:

\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})

Simplify

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

Add a constant to the solution

= - \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})

y = arcsin (8 x + 1)

What is the integral of (sin(x)-x)/(x^3) ?
The integral of (sin(x)-x)/(x^3) is -(sin(x)}{2x^2}+1/2 (-\frac{cos(x))/x-\Si(x))+1/x+C