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

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

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

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

- ln tan (\frac{1}{2 x}) + C

Solution steps

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

Apply u-substitution

= \int - csc (u) d u

Take the constant out:

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

= - \int csc (u) d u

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln tan (\frac{\frac{1}{x}}{2})

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= - ln tan (\frac{1}{2 x})

Add a constant to the solution

= - ln tan (\frac{1}{2 x}) + C

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