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

The integral of (cos(x)-(sin(x))/x)/(x) is (sin(x))/x+C

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integral of (cos(x)-(sin(x))/x)/(x)

\int \frac{cos ( x ) - \frac{s i n ( x )}{x}}{x} d x

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

Solution steps

\int \frac{cos ( x ) - \frac{s i n ( x )}{x}}{x} d x

Simplify

\frac{cos ( x ) - \frac{s i n ( x )}{x}}{x} : \frac{x cos ( x ) - sin ( x )}{x ^{2}}

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

Expand

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

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

\int \frac{cos ( x )}{x} d x = Ci (x)

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

= Ci (x) - (- \frac{sin ( x )}{x} + Ci (x))

Simplify

= \frac{sin ( x )}{x}

Add a constant to the solution

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

What is the integral of (cos(x)-(sin(x))/x)/(x) ?
The integral of (cos(x)-(sin(x))/x)/(x) is (sin(x))/x+C