Kumpulan Rumus-Rumus Dasar Integral

Pengertian integral

Jika $F(x)$ adalah fungsi umum yang bersifat $F'(x)= f(x)$, 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 $F'(x) =f(x)$
c         :  konstanta pengintegralan

Jenis integral

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

Sifat Umum Integral 

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

 Jika $f((x)$  dan $g(x)$ kontinue pada interval $a \leq x \leq b$, 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   $a \leq p \leq b$ 

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. $\displaystyle \int p^x dx = \frac{p^x}{ln p}+C$
   
   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. $\displaystyle \int p^{ax+b} dx = \frac{p^{ax+b}}{a.ln p}+C$
   
   d $\displaystyle \int \frac {1}{ax+b} dx = \frac{1}{a} ln^{|ax+b|}+C$
   
   
   

b. Integral Trigonometri

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