What is the integral of 1 (tan(x)) ?

The integral of 1 (tan(x)) is -dln|cos(x)|+C

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integral of 1 (tan(x))

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

- d ln ∣ cos (x) ∣ + C

Solution steps

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

Take the constant out:

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

= d \cdot \int tan (x) d x

Rewrite using trig identities

= d \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

= d \cdot \int - \frac{1}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= d (- ln ∣ u ∣)

Substitute back

u = cos (x)

= d (- ln ∣ cos (x) ∣)

Simplify

= - d ln ∣ cos (x) ∣

Add a constant to the solution

= - d ln ∣ cos (x) ∣ + C

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What is the integral of 1 (tan(x)) ?
The integral of 1 (tan(x)) is -dln|cos(x)|+C