Derivative and integral of arctan, with proofs

The inverse tangent has one of the neatest derivatives in calculus, and that single result unlocks its integral, its power series and a famous formula for π.

The derivative

Result: d/dx arctan x = 1 / (1 + x²)

Proof. Let y = arctan x, so tan y = x. Differentiate both sides with respect to x:

1. sec² y · dy/dx = 1
2. dy/dx = 1 / sec² y
3. Since sec² y = 1 + tan² y = 1 + x², we get dy/dx = 1 / (1 + x²).

With the chain rule: d/dx arctan(u) = u′ / (1 + u²). For example d/dx arctan(3x) = 3 / (1 + 9x²).

The integral of 1/(1 + x²)

Reading the derivative backwards gives one of the most used integrals:

∫ 1 / (1 + x²) dx = arctan x + C

More generally, ∫ 1 / (a² + x²) dx = (1/a) · arctan(x / a) + C.

The integral of arctan x

Integrate by parts with u = arctan x and dv = dx:

1. du = dx / (1 + x²), v = x
2. ∫ arctan x dx = x · arctan x − ∫ x / (1 + x²) dx
3. The remaining integral is ½ ln(1 + x²).

∫ arctan x dx = x · arctan x − ½ ln(1 + x²) + C

The Taylor series

For |x| < 1, 1 / (1 + x²) is a geometric series: 1 − x² + x⁴ − x⁶ + … Integrating term by term:

arctan x = x − x³/3 + x⁵/5 − x⁷/7 + …

The series also converges at x = ±1, although slowly.

The Leibniz formula for π

Put x = 1 into the series. Because arctan 1 = π/4:

π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − …

It is beautiful but converges very slowly. Faster formulas, such as Machin’s π/4 = 4 arctan(1/5) − arctan(1/239), were used to compute π by hand to over 100 digits.

Quick reference

Try it yourself: open the inverse tangent calculator to check any value from this article with full step-by-step working.