Integral of a Constant and of 1: Formula and Examples
The integral of a constant k is kx + C, and the integral of 1 is x + C. See why, the area picture, definite integrals and the mistakes to avoid.
Powers, roots and rational functions Updated 3 min read
At a glance
- Antiderivative
- kx + C
- Integral of 1
- \int 1\,dx = x + C
- Integral of 0
- \int 0\,dx = C
- Definite integral
- \int_a^b k\,dx = k(b - a)
- Check
- \dfrac{d}{dx}(kx + C) = k
On this page
The integral of a constant k is kx + C. In particular, the integral of 1 is x + C. It is the simplest integral in calculus, and almost every other integral uses it somewhere, usually as the last term of a polynomial.
Graph: a constant gives a rectangle
The area under a horizontal line is always a rectangle. Its height is k and its width is how far x has moved, so the accumulated area grows like kx. That is the whole formula in one picture.
Why the integral of k is kx + C
An integral asks: what function has this derivative? We want a function whose slope is k everywhere. A straight line with slope k does exactly that:
Adding any constant C moves the line up or down without changing its slope, so every function kx + C works. You can also see it as the reverse power rule with n = 0: since k = kx^0,
Check by differentiating
Always check an antiderivative by taking its derivative:
The matching derivative page is the derivative of a constant, which runs the same idea backwards: the derivative of any constant is 0.
Definite integrals of a constant
With the Fundamental Theorem of Calculus:
If k is negative, the rectangle sits below the axis and the answer is negative. For example, \int_0^4 (-2)\,dx = -8.
Worked examples
Example 1. \int 7\,dx = 7x + C.
Example 2: constants that look like variables. Any number without x counts as a constant, even if it is written with a symbol:
Example 3: inside a polynomial. Using the linearity rules:
The 9 becomes 9x. See the integral of x and the integral of x² for the other two terms.
Example 4: a different variable. \int 3\,dt = 3t + C. The variable in the answer matches the variable in dt.
Example 5: a definite integral. \int_{-1}^{4} 3\,dx = 3\big(4 - (-1)\big) = 15.
Common mistakes
- Integrating a constant to 0. That is the derivative rule. Integration goes the other way: \int 5\,dx = 5x + C, not 0.
- Writing 5 + C. The answer must contain x. Check: the derivative of 5 + C is 0, not 5.
- Forgetting + C. Without it you only have one antiderivative, not all of them.
- Treating e^2 or \pi^2 as a power of x. These are plain numbers. \int e^2\,dx is e^2x + C, not \frac{e^3}{3}.
Want more practice? Try the antiderivative practice problems or read the guide to antiderivatives.
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The reverse power rule
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