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A definite integral is an integral in which limits are applied to the integral.

A definite integral is usually represented by the following

$$\int^{b}_{a}f(x) dx$$

Here, $a$ is known as the lower limit and $b$ is the upper limit.

For example,  If we are to calculate the increase in the value of the integral $3x^2$ from 1 to 3, it will be represented as:

$$\int^{3}_{1}3x^2 dx$$

The above can be calculated thus:

$\int^{3}_{1}3x^2 dx=\left[\frac{3x^{2+1}}{3}+C\right]^3_1$





Notice that the arbitrary constant, C, cancels out indefinite integral.

This tells you that the arbitrary Constant, C is not always added to definite integral because it will cancel out when the limit of the integral is applied.

Example 1

Evaluate $\int^4_2(x^2+4) dx$


Using the power rule of integration



Substituting the upper and lower limit



Taking L.C.D


Example 2

Given that $f(x)=3x^2-8x+4$. Find the $\int^3_2 f(x) dx$


$\int^3_2 f(x) dx$ means we should find the definite integral of f(x) from 2 to 3

$\int^3_2 (3x^2-8x+4) dx=\left[\frac{3x^{2+1}}{3}-\frac{8x^{1+1}}{2}+\frac{4x^{0+1}}{1}\right]^3_2$.


By substitution




Example 3

Evaluate $\int^2_14e^{2x}$


Recalled that $\int e^{ax}=\frac{e^{ax}}{a}$



Applying the definite integral



Example 4

Evaluate $\int^2_0 3\sin x dx$


Since the derivative of $\cos x$ is $-\sin x$, it follows that the integral of $\sin x$ is $-\cos x$

$\left[-3 \cos x\right]^2_0$

$(-3 \cos 2)-(-3 \cos 0)$


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