What is the integral of atan(x) ?

The integral of atan(x) is -aln|cos(x)|+C

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integral of atan(x)

\int a tan (x) d x

- a ln ∣ cos (x) ∣ + C

Solution steps

\int a tan (x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

= a \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a (- ln ∣ u ∣)

Substitute back

u = cos (x)

= a (- ln ∣ cos (x) ∣)

Simplify

= - a ln ∣ cos (x) ∣

Add a constant to the solution

= - a ln ∣ cos (x) ∣ + C

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