What is the integral of x(1/(x^2)-(sin(x))/x) ?

The integral of x(1/(x^2)-(sin(x))/x) is ln|x|+cos(x)+C

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integral of x(1/(x^2)-(sin(x))/x)

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

ln ∣ x ∣ + cos (x) + C

Solution steps

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

Simplify

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int sin (x) d x = - cos (x)

= ln ∣ x ∣ - (- cos (x))

Simplify

= ln ∣ x ∣ + cos (x)

Add a constant to the solution

= ln ∣ x ∣ + cos (x) + C

What is the integral of x(1/(x^2)-(sin(x))/x) ?
The integral of x(1/(x^2)-(sin(x))/x) is ln|x|+cos(x)+C