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

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

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integral of 1/(x^2)tan(1/x)

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

ln cos (\frac{1}{x}) + C

Solution steps

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

Simplify

\frac{1}{x ^{2}} tan (\frac{1}{x}) : \frac{tan ( \frac{1}{x} )}{x ^{2}}

= \int \frac{tan ( \frac{1}{x} )}{x ^{2}} d x

Apply u-substitution

: \int - tan (u) d u

= \int - tan (u) d u

Take the constant out:

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

= - \int tan (u) d u

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{v} d v

= - \int - \frac{1}{v} d v

Take the constant out:

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

= - (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - (- ln ∣ v ∣)

Substitute back

= - (- ln cos (\frac{1}{x}))

Simplify

= ln cos (\frac{1}{x})

Add a constant to the solution

= ln cos (\frac{1}{x}) + C

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