What is the integral of 4/(x^2)sec(1/x)tan(1/x) ?

The integral of 4/(x^2)sec(1/x)tan(1/x) is -4sec(1/x)+C

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integral of 4/(x^2)sec(1/x)tan(1/x)

\int \frac{4}{x ^{2}} sec (\frac{1}{x}) tan (\frac{1}{x}) d x

- 4 sec (\frac{1}{x}) + C

Solution steps

\int \frac{4}{x ^{2}} sec (\frac{1}{x}) tan (\frac{1}{x}) d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{x ^{2}} sec (\frac{1}{x}) tan (\frac{1}{x}) d x

Apply u-substitution

: \int - sec (u) tan (u) d u

= 4 \cdot \int - sec (u) tan (u) d u

Take the constant out:

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

= 4 (- \int sec (u) tan (u) d u)

Apply u-substitution

: \int 1 d v

= 4 (- \int 1 d v)

Integral of a constant:

\int a d x = a x

= 4 (- 1 \cdot v)

Substitute back

= 4 (- 1 \cdot sec (\frac{1}{x}))

Simplify

= - 4 sec (\frac{1}{x})

Add a constant to the solution

= - 4 sec (\frac{1}{x}) + C

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What is the integral of 4/(x^2)sec(1/x)tan(1/x) ?
The integral of 4/(x^2)sec(1/x)tan(1/x) is -4sec(1/x)+C