What is the integral of 1/x cos(1/x) ?

The integral of 1/x cos(1/x) is -\Ci(1/x)+C

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integral of 1/x cos(1/x)

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

- Ci (\frac{1}{x}) + C

Solution steps

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

Simplify

\frac{1}{x} cos (\frac{1}{x}) : \frac{cos ( \frac{1}{x} )}{x}

= \int \frac{cos ( \frac{1}{x} )}{x} d x

Apply u-substitution

: \int - \frac{cos ( u )}{u} d u

= \int - \frac{cos ( u )}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{cos ( u )}{u} d u

Use the nonelementary integral:

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

= - Ci (u)

Substitute back

u = \frac{1}{x}

= - Ci (\frac{1}{x})

Add a constant to the solution

= - Ci (\frac{1}{x}) + C

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What is the integral of 1/x cos(1/x) ?
The integral of 1/x cos(1/x) is -\Ci(1/x)+C