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Primitives

The primitive of a function $$f$$ is a differentiable function $$F$$ such that $$$ F' = f.$$$ The function $$F$$ is also called the anti-derivative or indefinite integral. It is often denoted similarly to the integral, with no upper or lower limits $$$ \int f(x)\,dx.$$$

$$\sin x$$, $$\sin x + 1$$, $$\sin x - \pi$$ are primitives of $$\cos x$$.

On a given interval, a continuous function has an infinite number of primitives. More precisely, all primitives differ by a constant.

All the primitives of $$3x^2$$ are of the form $$x^3 + c$$ for any constant $$c$$.

Proof. If $$F_1$$ and $$F_2$$ are two primitives of the same function, the function $$F=F_1-F_2$$ satisfies $$F'=0$$; it is therefore a constant.

Primitives of the same function.
Primitives of the same function.

The fundamental theorem of calculus connects integration to differentiation. This is why integration is also called anti-differentiation.

The first fundamental theorem of calculus states that $$$ F(x) = \int_a^x f(t)\,dt$$$ is a primitive of $$f$$. It is the unique primitive of $$f$$ such that $$F(a) = 0$$.

Proof We have that $$$\frac{F(x) - F(x_0)}{x-x_0} - f(x_0) = \frac{1}{(x-x_0)}\int_{x_0}^{x}\big(f(t) - f(x_0)\big)\,dt $$$ The right-hand term tends to 0 when $$x\to x_0$$ since $$f(x)\to f(x_0)$$. Thus, $$F$$ is differentiable at $$x_0$$ and its derivative is $$f(x_0)$$.

The first theorem of calculus
The first theorem of calculus

The second fundamental theorem of calculus states that, for a differentiable function, $$\displaystyle \int_a^b f'(t)\,dt = f(b) - f(a).$$

In other words, if $$F$$ is a primitive of $$f$$, we have $$$ \int_a^b f(t)\,dt = F(b) - F(a) .$$$

Proof We set $$\displaystyle g(x) = \int_a^x f'(t)\,dt.$$ By the first fundamental theorem, $$g' = f'$$ on $$(a,b)$$. So, $$g-f$$ is constant. We conclude that $$$\int_a^b f'(t)\,dt = g(b) - g(a) = f(b)-f(a).$$$

The second theorem of calculus
The second theorem of calculus

Integration is a linear operator. For two functions and two numbers, $$$ \int_a^b\big(\lambda f(x)+\mu g(x)\big)\,dx = \lambda \int_a^b f(x)\,dx + \mu \int_a^b g(x)\,dx. $$$

  • Reversing the limits of integration: By convention, $$$ \int_b^a f(x)\,dx = - \int_a^b f(x)\,dx $$$
  • Integral with identical limits $$$ \int_a^a f(x)\,dx = 0 $$$
  • Addition of integration on intervals: (Chasles relation) $$$ \int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx $$$

These properties are immediate when stated with a primitive $$F$$ of $$f$$. For instance, addition of integration on intervals reads $$$ F(c) - F(a) = F(b) - F(a) + F(c) - F(b).$$$

Addition of integration on intervals
Addition of integration on intervals