Kumpulan Rumus-Rumus Dasar Integral

Pengertian integral

Jika $F(x)$ adalah fungsi umum yang bersifat $F^{\prime }(x)=f(x)$, maka $F(x)$ merupakan kebalikan dari turunan atau antiturunan. Pengintegralan fungsi $f(x)$ terhadap x dinotasikan sebagai berikut:

                                                   $\int f(x)dx=F(x)+C$

$\int dx$    : notasi integral (yang diperkenalkan oleh Leibniz, seorang matematikawan Jerman)
$f(x)$    :  fungsi integran
$F(x)$   : fungsi integral umum yang bersifat $F^{\prime }(x)=f(x)$
c         :  konstanta pengintegralan

Jenis integral

1. Integral tak tentu                      $\int f(x)dx=F(x)+C$
   
   
2. Integral tentu                            ${\displaystyle {\int }_a^{b}f(x)dx={[F(x)+C]}_a^b=F(b)-F(a)}$   dengan $a\le x\le b$

Sifat Umum Integral 

1. $\int cf(x)dx=c\int f(x)dx$   dengan c adalah konstanta
   
   
2. $\int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx$
   
   

 Jika $f((x)$  dan $g(x)$ kontinue pada interval $a\le x\le b$, maka 

1. ${\int }_a^{a}f(x)dx=0$
   
   
2. ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
   
   
3. ${\int }_a^{b}f(x)dx={\int }_a^{m}f(x)dx+{\int }_m^{b}f(x)dx$   dimana   $a\le p\le b$ 

Rumus Integral

a. Integral fungsi aljabar

1. Rumus dasar
   
   a. ${\displaystyle \int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C}$
   
   b. ${\displaystyle \int e^{n}dx=e^x+C}$
   
   c. ${\displaystyle \int p^{x}dx=\frac{p^x}{lnp}+C}$
   
   d. ${\displaystyle \int \frac{1}{x}dx=ln|x|+C}$
   
    
2. Rumus Pengembangan
   
   a. ${\displaystyle \int (ax+b{)}^{n}dx=\frac{1}{a(n+1)}(ax+b{)}^{n+1}+C}$
   
   b. ${\displaystyle \int e^{ax+b}dx=\frac{1}{a}{e}^{ax+b}+C}$
   
   c. ${\displaystyle \int p^{ax+b}dx=\frac{p^{ax+b}}{a.lnp}+C}$
   
   d ${\displaystyle \int \frac{1}{ax+b}dx=\frac{1}{a}l{n}^{|ax+b|}+C}$
   
   
   

b. Integral Trigonometri

1. Rumus dasar
   
   a. ${\displaystyle \int \sin xdx=-\cos x+C}$
   
   b. ${\displaystyle \int \cos xdx=\sin x+C}$
   
   c. ${\displaystyle \int \tan xdx=-ln|\cos x|+C}$
   
   d. ${\displaystyle \int \cot xdx=ln|\sin x|+C}$
   
   e. ${\displaystyle \int (\csc x{)}^{2}dx=-\cot x+C}$
   
   f. ${\displaystyle \int (\cot x{)}^{2}dx=\tan x+C}$
   
   g. ${\displaystyle \int \tan x.\sec xdx=\sec x+C}$
   
   h. ${\displaystyle \int \cot x.\csc xdx=-\csc x+C}$
   
   
   
2. Rumus pengembangan
   
   a. ${\displaystyle \int \sin (ax+b)dx=-\frac{1}{a}\cos (ax+b)+C}$
   
   b. ${\displaystyle \int \cos (ax+b)dx=\frac{1}{a}\sin (ax+b)+C}$
   
   c. ${\displaystyle \int \tan (ax+b)dx=-\frac{1}{a}|\cos (ax+b)|+C}$
   
   d. ${\displaystyle \int \cot (ax+b)dx=\frac{1}{a}|\sin (ax+b)|+C}$
   
   
e. ${\displaystyle \int (\csc x{)}^{2}dx=-\frac{1}{a}|\cot (ax+b)|+C}$

f. ${\displaystyle \int (\sec x{)}^{2}dx=\frac{1}{a}|\tan (ax+b)|+C}$

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