Integral

Vocabulary of the equation

1. c = constant
2. n = constant
3. $\int _{a}^{b}$ = Integrals from a to b

Definite Integral

Some equations you can remember But when you are looking at the equation you must need to know F(x)= f'(x).

1. $\int _{a}^{b}f(x)\,dx=F(b)-F(a)$
2. $\int _{a}^{b}c{x^{n}}\,dx=c{b^{n+1} \over n+1}-c{a^{n+1} \over n+1}$

Indefinite Integral

Some equations you can remember But same you must need to know F(x)= f'(x).

1. Indefinite Integral $\int f(x)\,dx=F(x)+c,$
2. sum rule of Indefinite Integral $\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$
3. The Difference Rule $\int [f(x)-g(x)]\,dx=\int f(x)\,dx-\int g(x)\,dx$
4. Indefinite Integral $\int x^{n}\,dx={x^{n+1} \over n+1}+c$
5. Natural log rule $\int {n \over x}\,dx={ln|x^{n}|}$
6. $\int a^{x}dx={a^{x} \over ln(a)}$
7. constant(constant can be pull out in the Indefinite Integral) $\int c*f(x)dx=c\int f(x)dx$

Examples

Examples for Definite Integral

1. $\int _{a}^{b}f(x)\,dx=F(b)-F(a)$

This is the rule that we will going to used for calculating areas

Know Here are some example for you to do

Examples for Definite Integral 1

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx$

using this equation $\int _{a}^{b}c{x^{n}}\,dx=c{b^{n+1} \over n+1}-c{a^{n+1} \over n+1}$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(8({3^{3+1} \over 3+1})+3({3^{2+1} \over 2+1})+6({2^{1+1} \over 1+1}))-(8({2^{3+1} \over 3+1})+3({2^{2+1} \over 2+1})+6({2^{1+1} \over 1+1}))$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(8({3^{4} \over 4})+3({3^{3} \over 3})+6({3^{2} \over 2}))-(8({2^{4} \over 4})+3({2^{3} \over 3})+6({2^{2} \over 2}))$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(8({81 \over 4})+3({27 \over 3})+6({9 \over 2}))-(8({16 \over 4})+3({8 \over 3})+6({4 \over 2}))$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(2(81)+(27)+3(9))-(2(16)+(8)+3(4))$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(162+27+27)-(32+8+12))$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=(216)-(52)$

$\int _{2}^{3}8x^{3}+3x^{2}+6x\,dx=164$