Differentials and integrals: Area and primitive function
General definition and basic rules of integrals
Let \(F(x)\) be a primitive function of \(f(x)\) on an interval \(I\). For \(a\lt b \) in \(I\), the difference between \(F(b)-F(a)\) is equal to the area of a plane region and independent of the choice of the primitive function \(F(x)\). This difference is also called the definite integral of \(f(x)\) with lower bound \(a\) and upper bound \(b\) denoted as \(\int_a^b f(x)\,\dd x,\) and can also be calculated in case \(a\ge b\) .
From the general integral definition immediately follow the below properties:
Properties of integrals
- \(\displaystyle\quad \int_a^b f(x)\,\dd x = -\int_b^a f(x)\,\dd x\)
- \(\displaystyle\quad \int_a^b c\cdot f(x)\,\dd x = c\cdot \int_a^b f(x)\,\dd x\quad\) for any constant \(c\).
- \(\displaystyle\quad \int_a^b \bigl(f(x)+g(x)\bigr)\,\dd x = \int_a^b f(x)\,\dd x +\int_a^b g(x)\,\dd x\)
- \(\displaystyle\quad \int_a^c f(x)\,\dd x =\int_a^b f(x)\,\dd x + \int_b^c f(x)\,\dd x\quad\) for all \(a,b,c\) in \(I\).
- \(\displaystyle\quad \frac{d}{\dd x}\left(\int_a^x f(\xi)\,d\xi\right) = f(x)\quad \text{and}\quad \frac{d}{\dd x}\left(\int_x^b f(\xi)\,d\xi\right) = -f(x)\)
- \(\displaystyle\quad \int_a^b f(x)\,\dd x = c\cdot \int_{a/c}^{b/c} f(c\cdot x)\,\dd x\quad\) for any constant \(c\neq0\).
