Integrals and primitives: senior math course in PDF.

Integrals and primitives with a senior math course free download PDF.

In this lesson we will see the definition and the different properties of the integral as well as the geometrical meaning with the areas. In addition, in this course you will see the different ways to calculate an integral using the primitive and the associativity and linearity properties of the integral. In addition,integrals and primitives must be mastered in order to progress well in math .

I. Definitions and properties of the integral and primitives.

A function f is integrable on an interval I when it has primitives on this interval I.
In particular, functions which are derivable on an interval I, are integrable on I, but this condition, although sufficient, is not necessary. The function f defined on \mathbb{R} by f(x)=2%7Cx%7C, although not derivable at 0, is integrable on \mathbb{R}, because it has as primitive on \mathbb{R} the function F such that: F(x)\,=\,x\,%7C\,x\,%7C.
However, except in special cases containing indications, in the baccalaureate problems, functions integrable on an interval I will always be functions derivable on I.
If F and G are two primitives of an integrable function on an interval I,
then whatever a \in I and b \in I, we have
F(b) – F(a) = G(b) – G(a)Indeed, if F ‘(x) = G ‘(x), then there exists c \in \mathbb{R} such that G(x) = F(x) + c.
Therefore: G(b) – G(a) = F(b) + c – [F(a) + c] = F(b) – F(a).


The difference in the images of b and a for any primitive of f is the same.

This number depends only on f, a and b.

This will allow us to give the following definition:

Let f be an integrable function on an interval I,
Let a \in I and b \in I be two real numbers of this interval I,
The integral from a to b of the function f is the number F(b) – F(a) where F is any primitive of f on I.


F(b)\,-\,F(a)\,=\,\int_{a}^{b}f(t)dt which reads“sum from a to b of f from t dt ” and is also called“integral of f between a and b“.

In the writing \int_{a}^{b}f(t)dt, the letter t is called: “dumb variable”.
Indeed, we can also write \int_{a}^{b}f(t)dt=\int_{a}^{b}f(u)du=\int_{a}^{b}f(x)dxThe letter “dummy variable” indicates the name of the “variable of integration” (any other letter in the expression of the function f to be integrated is then considered as constant. The interest of this appears when there are several variables, but this is not in the Terminale program, however, it will be useful in the presence of parameters).

\int_{a}^{b}f(t)dt is also written in the condensed form using F :



The function f:x\,\mapsto  \,x^2 is derivable on \mathbb{R}.

It is therefore integrable on \mathbb{R} and admits primitives on \mathbb{R}.

For example F:x\,\mapsto  \,\frac{x^3}{3} is a primitive of f on \mathbb{R}.

So we have:

II. consequences of the definitionThe first properties.


Let f be a function derivable on an interval [a,b].\int_{a}^{a}f(t)dt=0


If f is derivable on I, if a \in I and b \in I, we havef(b)-f(a)=\int_{a}^{b}f'(t)dt
From this property, we deduce an important synthesis that links a derivable function, its derivative function and the notion of integral.
If f is a differentiable function on I and if a \in I.
Then, for all x \in I, we have f(x)\,=\,f(a)\,+\int_{a}^{x}f'(t)dt.

III.primitive function of an integrable function.

If f is an integrable function on an interval I and if a \in I, then
the function F_a defined on I by F_a(x)\,=\,\int_{a}^{x}f\,(t)dt is the primitive of f which cancels at x = a.

If G is any primitive of f on I, then F_a(x)\,=\,\int_{a}^{x}f\,(t)dt=G(x)-G(a), so
Fa‘(x) = G ‘(x) = f(x) .

Indeed, G(a) is a constant, so its derivative is zero and G ‘(x) = f(x) because G is a primitive of f. Conclusion: F_a is also a primitive of f .
Moreover:Fa (a) = G(a) – G(a )= 0, soFa cancels for x=a .


Compute the following integrals, then state the primitives they define.
F(x)=\int_{0}^{x}(2t)dt G(x)=\int_{1}^{x}(3t^2)dt H(x)=\int_{-1}^{x}(\,3t^2+\,2t+1)dt

IV. Integral of a positive function.

We assume here that f is integrable and positive on [a;b], that is to say that for all x \in [a;b], we have: f(x) \geq\, 0.
Call F a primitive of f on [a;b]Then we have F ‘(x) = f(x) for all x \in [a;b]The derivative f of F being positive on [a;b]the function F is increasing on [a;b].as a < b, then we have: F(a) \leq\, F(b).
So: \int_{a}^{b}\,f(t)dt=F(b)-F(a)\geq\,\,0.
If a < b and if for all x \in [a;b] we have f(x)\geq\, 0 then \int_{a}^{b}\,f(t)dt\,\geq\,\,0.

V. Integrals and areas

1. summary table

integral and area

In an orthogonal frame of reference (O,\vec{i},\vec{j}), we represent graphically a function f derivable and positive on an interval [a;b]. We call D the set of points of the plane whose coordinates (x;y) verify :
a \leq\, x \leq\, b and 0 \leq\, y \leq\, f(x)Then Area of D = \int_{a}^{b}f(t)dt units of areaThe unit of area being that of the rectangle whose sides are the units of length of the abscissa and ordinate.
The set D consists of the points located between the curve representing the function f, the x-axis and the lines of equation x = a and x = b.

integral area

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