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
- Integral tak tentu $\int f(x) dx = F(x)+C$
- 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
- $\int c f(x)dx=c \int f(x)dx$ dengan c adalah konstanta
- $\int \left ( f(x) \pm g(x) \right ) dx = \int f(x)dx \pm \int g(x)dx $
- $\int_{a}^{a} f(x)dx = 0$
- $\int_{a}^{b} f(x)dx = - \int_{b}^{a} f(x)dx$
- $\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
- 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. $\displaystyle \int \frac {1}{x} dx = ln |x|+C$
- 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
- 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. $\displaystyle \int \cot {x}.\csc{x} dx = -\csc {x}+C$ - 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. $\displaystyle \int (\sec{x})^2 dx = \frac{1}{a} |\tan {(ax+b)}|+C$
f. $\displaystyle \int (\sec{x})^2 dx = \frac{1}{a} |\tan {(ax+b)}|+C$
