Definite Intergral Basics

Prerequisite

• Derivatives
• AntiDifferentiation
• Indefinite Integral

Definite Integral

Definite Integral represents area between the function and x axis.

If the area is above the x axis, the result will be positive.

If the area is under the x axis, the result will be negative.

In other words, it means the total change of $F(x)$ from $a$ to $b$.

So if the derivative is at negative value, the fucntion $F(x)$ will decrease, and it is the reason why the integral or the area under x axis is negative.

\[\begin{gather} \int_a^b f(x) \space dx = F(b) - F(a) \\ \text{where } f(x) = \frac{d}{dx}F(x) \end{gather}\]

When rearranged, we get

\[\begin{gather} F(b) = F(a) + \int_a^b f(x) \space dx \end{gather}\]

which means the value of $F(b)$ is $F(a)$ plus the accumulation of change from $a$ to $b$.

Proporties of Definite Integral

\[\begin{align} &(1) \quad \int_a^b cf(x) \space dx = c \int_a^b f(x) \space dx \\ &(2) \quad \int_a^b f(x) \pm g(x) \space dx = \int_a^b f(x) \space dx \pm \int_a^b g(x) \space dx \\ &(3) \quad \int_a^b c \space dx = c(b-a) \\ &(4) \quad \int_a^a f(x) \space dx = 0 \\ &(5) \quad \int_a^b f(x) \space dx = - \int_b^a f(x) \space dx \\ \end{align}\]

Differentiating the integral

\[\begin{align} \frac{d}{dx} \left[ \int_a^x f(t) \space dt \right] &= \frac{d}{dx}F(x) - \frac{d}{dx}F(a) \\ &= f(x) - 0 \\ &= f(x) \end{align}\]

Example

\[\begin{align} \frac{d}{dx} \left[ \int_0^{2x} \ln t^2 \space dt \right] &= \frac{d}{dx}F(2x) - \frac{d}{dx}F(0) \\ &= F'(2x) \cdot \frac{d}{dx}2x - 0 \\ &= 2f(2x) \\ &= 2 \ln (2x)^2 \\ &= 2 \ln 4x^2 \end{align}\]