What is the integral of artan(x) ?

The integral of artan(x) is -arln|cos(x)|+C

integral of artan(x)

\int a r tan (x) d x

- a r ln ∣ cos (x) ∣ + C

Solution steps

\int a r tan (x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a r (- ln ∣ u ∣)

Substitute back

u = cos (x)

= a r (- ln ∣ cos (x) ∣)

Simplify

= - a r ln ∣ cos (x) ∣

Add a constant to the solution

= - a r ln ∣ cos (x) ∣ + C

\int e^{- x} \frac{1}{x} d x

\int 2 cos^{3} (2 x) d x

d er i v a t i v e sin (θ) + csc (θ)

\frac{d}{d x} (\frac{a ^{2} + x ^{2}}{a ^{2} - x ^{2}})

\int_{- 1}^{0} 2 - e^{- x} d x