Integration Calculus

Integration is a mathematical concept of adding segements to find the final result, it is also known has a reserve of differetiation. the result from an Integration is an integral

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How to Integrate

Equation: y = ${\mathrm{ax}}^3$

1. Add one (1) to the power. In this case: ${\mathrm{ax}}^{\mathrm{3\; +\; 1}}$

2. Divide the cooefficient by the new power. In this case:

3. add C to the equation which is the constant . In this case

4. Merge the steps together.

Point to note

1. when integrating a number the result contains a variable which is added to the number

The Integration of the differetiated equation above is found by the following steps

Given this equation $\mathrm{y}={\mathrm{2x}}^3-{\mathrm{3x}}^2-\mathrm{36x}-3$ prove by reversing the differetiated equation $\frac{\mathrm{dy}}{\mathrm{dx}}$ back this equation

The differetiated equation $\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{6x}}^2-\mathrm{6x}-36$

Solution:

$\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{6x}}^2-\mathrm{6x}-36$

$\int ({\mathrm{6x}}^2-\mathrm{6x}-36)\mathrm{dx}$

$\int (\frac{{\mathrm{6x}}^{\mathrm{2\; +\; 1}}}{3}-\frac{{\mathrm{6x}}^{\mathrm{1\; +\; 1}}}{2}-\mathrm{36x}+\mathrm{C})\mathrm{dx}$

$\int (\frac{{\mathrm{6x}}^3}{3}-\frac{{\mathrm{6x}}^2}{2}-\mathrm{36x}+\mathrm{C})\mathrm{dx}$

$\int ({\mathrm{2x}}^3-{\mathrm{3x}}^2-\mathrm{36x}+\mathrm{C})\mathrm{dx}$

Integral : $\int ({\mathrm{2x}}^3-{\mathrm{3x}}^2-\mathrm{36x}+\mathrm{C})\mathrm{dx}$

C is -3 which is the constant of the equation

Example 1

Integrate the following equations with respect to x

(a) $y={\mathrm{20x}}^4$

(b) $y=\mathrm{2x}$

(c) $y={\mathrm{12x}}^3+19$

Solutions

(a)

$y={\mathrm{20x}}^4$

$\int ={\mathrm{20x}}^4$

$\int ={\frac{\mathrm{20x}}{5}}^{\mathrm{4\; +\; 1}}$

$\int ={\frac{\mathrm{20x}}{5}}^5$

Answer: $\int ={\mathrm{4x}}^5$ + C

or $\int ={\mathrm{4x}}^5$

(b)

$y=\mathrm{2x}$

$\int =\mathrm{2x}$

$\int ={\frac{\mathrm{2x}}{2}}^{\mathrm{1\; +\; 1}}$

$\int ={\frac{\mathrm{2x}}{2}}^2$

Answer: $\int =x^2$ + C

or $\int =x^2$

(c)

$y={\mathrm{12x}}^3+19$

$\int ={\mathrm{12x}}^3+19$

$\int ={\frac{\mathrm{12x}}{4}}^{\mathrm{3\; +\; 1}}+\mathrm{19x}$

$\int ={\frac{\mathrm{12x}}{4}}^4+\mathrm{19x}$

Answer: $\int ={\mathrm{3x}}^4+\mathrm{19x}$ + C

or $\int ={\mathrm{3x}}^4+\mathrm{19x}$

Example 2

Evaluate $\underset{-1}{\overset{\mathrm{2\; <!--INFINITY-->}}{\int \; <!--INTEGRAL-->}}(2+\mathrm{x}-x^2)\mathrm{dx}$

$[\mathrm{2x}+{\frac{x}{2}}^2-{\frac{x}{3}}^3]$

$[\mathrm{2(2)}+{\frac{2}{2}}^2-{\frac{2}{3}}^3]$ - $[\mathrm{2(-1)}+{\frac{\mathrm{-1}}{2}}^2-{\frac{\mathrm{-1}}{3}}^3]$

$[4+2-\frac{8}{3}]$ - $[-2+\frac{1}{2}-\frac{\mathrm{-1}}{3}]$

$[6-2.67]$ - $[-2+\mathrm{0.5}+0.33]$

$\left[3.33\right]$ - $\left[-1.17\right]$

$\left[3.33\right]$ + $\left[1.17\right]$

Answers: $4.5$

How to obtaining an integral from a derivation

Integral with limits