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# Primitives

## Primitive of a function

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.

## The first fundamental theorem of calculus

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 second fundamental 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

## Operations on the limits of an integral

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