# Arctangent arctan(x): Graph, Asymptotes, Derivative and Integral

> The arctangent function arctan(x), the inverse of tangent: domain all reals, range (−π/2, π/2), graph, derivative 1/(1+x²), integral and examples.

Source: https://practicecalculus.com/functions/arctangent
Updated: 2026-11-03

$$ f(x) = \arctan x = \tan^{-1} x $$

## At a glance

- **Domain:** $(-\infty, \infty)$
- **Range:** $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
- **Derivative:** $\frac{1}{1 + x^2}$
- **Integral:** $x\arctan x - \frac{1}{2}\ln(1 + x^2) + C$
- **Horizontal asymptotes:** $y = \pm\frac{\pi}{2}$
- **Symmetry:** Odd: $\arctan(-x) = -\arctan x$

## Key takeaways

- $\arctan x$ is the angle in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ whose tangent is $x$, defined for every real $x$.
- Its derivative is $\frac{1}{1+x^2}$, so $\int\frac{dx}{1+x^2} = \arctan x + C$.
- The graph has horizontal asymptotes $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$.

The **arctangent function** undoes tangent: $\arctan x$ is the angle in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ whose tangent is $x$. Unlike [arcsine](/functions/arcsine) and [arccosine](/functions/arccosine), it accepts every real number, because [tangent](/functions/tangent-function) takes every real value. In calculus it is famous as the antiderivative of $\frac{1}{1+x^2}$.

## Definition of arctan x

$$y = \arctan x \quad\Longleftrightarrow\quad \tan y = x \text{ and } -\frac{\pi}{2} < y < \frac{\pi}{2}$$

Since $\tan y$ is the slope of a line at angle $y$, you can think of $\arctan m$ as the angle a line of slope $m$ makes with the positive $x$-axis. For example $\arctan 1 = \frac{\pi}{4}$: a line of slope $1$ makes a 45 degree angle.

## Graph of y = arctan x

```graph
title: 'Graph of y = arctan x with horizontal asymptotes y = ±π/2'
caption: 'Arctangent increases everywhere and levels off toward $\frac{\pi}{2}$ on the right and $-\frac{\pi}{2}$ on the left.'
x: [-8, 8]
y: [-2.2, 2.2]
functions:
  - expr: arctan(x)
    label: y = arctan x
  - expr: '1.5708'
    label: y = π/2
    dashed: true
  - expr: '-1.5708'
    label: y = −π/2
    dashed: true
points:
  - [0, 0, '(0, 0)']
  - [1, 0.7854, '(1, π/4)']
```

## Key values of arctangent

| $x$ | $-\sqrt{3}$ | $-1$ | $-\frac{\sqrt{3}}{3}$ | $0$ | $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $\arctan x$ | $-\frac{\pi}{3}$ | $-\frac{\pi}{4}$ | $-\frac{\pi}{6}$ | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ |

## Properties of arctangent

- **Domain all reals, range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.** The endpoints are never reached.
- **Increasing and odd.** $\arctan(-x) = -\arctan x$.
- **Limits at infinity.** $\lim_{x\to\infty}\arctan x = \frac{\pi}{2}$ and $\lim_{x\to-\infty}\arctan x = -\frac{\pi}{2}$. See [limits at infinity](/blog/limits-at-infinity).
- **Cancellation.** $\tan(\arctan x) = x$ for all $x$; $\arctan(\tan x) = x$ only for $x$ in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
- **Reciprocal identity.** For $x > 0$, $\arctan x + \arctan\frac{1}{x} = \frac{\pi}{2}$.

## Derivative of arctan x

$$\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$$

**Why.** Let $y = \arctan x$, so $\tan y = x$. [Implicit differentiation](/rules/implicit-differentiation) gives $\sec^2 y\cdot y' = 1$, so $y' = \frac{1}{\sec^2 y} = \frac{1}{1 + \tan^2 y} = \frac{1}{1 + x^2}$.

The derivative is always positive and largest at $x = 0$, where it equals $1$. It shrinks toward $0$ as $|x|$ grows, matching how the graph flattens.

## Integrals involving arctangent

$$\int\frac{dx}{1 + x^2} = \arctan x + C, \qquad \int\frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C \ (a > 0)$$

These appear whenever an irreducible quadratic sits in a denominator, which is common in [partial fractions](/blog/partial-fractions).

The integral of arctangent itself uses [integration by parts](/blog/integration-by-parts) with $u = \arctan x$, $dv = dx$:

$$\int\arctan x\,dx = x\arctan x - \int\frac{x}{1+x^2}\,dx = x\arctan x - \frac{1}{2}\ln(1 + x^2) + C$$

No absolute value is needed because $1 + x^2 > 0$.

## Worked examples

**Example 1: chain rule.** Differentiate $f(x) = \arctan(2x)$:

$$f'(x) = \frac{2}{1 + 4x^2}$$

**Example 2: an improper integral.**

$$\int_0^\infty\frac{dx}{1 + x^2} = \lim_{b\to\infty}\left(\arctan b - \arctan 0\right) = \frac{\pi}{2}$$

This is a classic convergent [improper integral](/blog/improper-integrals).

**Example 3: completing the square.** Evaluate $\int\frac{dx}{x^2 + 2x + 5}$. Write $x^2 + 2x + 5 = (x+1)^2 + 4$ and use $a = 2$:

$$\int\frac{dx}{(x+1)^2 + 4} = \frac{1}{2}\arctan\frac{x+1}{2} + C$$

**Example 4: a definite integral.** $\int_0^1\frac{dx}{1+x^2} = \arctan 1 - \arctan 0 = \frac{\pi}{4}$.

## Common mistakes

1. **Writing the derivative as $\frac{1}{\sqrt{1+x^2}}$.** There is no square root for arctangent. The square root belongs to arcsine and arccosine (with $1 - x^2$).
2. **Forgetting the $\frac{1}{a}$.** $\int\frac{dx}{9 + x^2} = \frac{1}{3}\arctan\frac{x}{3} + C$, not $\arctan\frac{x}{3} + C$.
3. **Reading $\tan^{-1}x$ as $\cot x$.** $\frac{1}{\tan x}$ is cotangent, a different function (see [reciprocal trig functions](/functions/reciprocal-trig-functions)).
4. **Answering outside the range.** $\arctan(-1) = -\frac{\pi}{4}$, not $\frac{3\pi}{4}$.

## How arctangent connects to other topics

The substitution $x = a\tan\theta$ in [trig substitution](/blog/trig-substitution) produces arctangent answers. Its [power series](/blog/power-series), $\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ for $|x| \le 1$, comes from integrating the [geometric series](/blog/geometric-series) for $\frac{1}{1+x^2}$, and at $x = 1$ it gives the famous $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \cdots$.

## Frequently asked questions

### What is the derivative of arctan x?

The derivative of $\arctan x$ is $\frac{1}{1+x^2}$. With the chain rule, $\frac{d}{dx}\arctan(u) = \frac{u'}{1+u^2}$.

### What is the integral of 1/(1+x^2)?

It is $\arctan x + C$. More generally, $\int\frac{dx}{a^2+x^2} = \frac{1}{a}\arctan\frac{x}{a} + C$ for $a > 0$.

### What is the limit of arctan x as x goes to infinity?

$\lim_{x\to\infty}\arctan x = \frac{\pi}{2}$ and $\lim_{x\to-\infty}\arctan x = -\frac{\pi}{2}$. These are the horizontal asymptotes of the graph.

### What is the integral of arctan x?

Integration by parts with $u = \arctan x$ and $dv = dx$ gives $\int\arctan x\,dx = x\arctan x - \frac{1}{2}\ln(1+x^2) + C$.