BELİRLİ İNTEGRALİN ÖZELLİKLERİ

1. f ve g fonksiyonları, [a,b] aralığında integrallenebilir iki fonksiyon ve a,b,c R ise;

\[\int\limits_a^b {[f(x) \mp g(x)]dx = \int\limits_a^b {f(x)dx \mp \int\limits_a^b {g(x)dx} } } \]

\[\int\limits_{\pi /2}^\pi  {(3.\sin x + \cos x)dx}  = \int\limits_{\pi /2}^\pi  {3.\sin xdx}  + \int\limits_{\pi /2}^\pi  {\cos xdx}  = 3( – \cos x)\mathop {{\text{ }}|}\limits_{\pi /2}^\pi   + \sin x\mathop {{\text{ }}|}\limits_{\pi /2}^\pi  \]

\[ – 3((\cos \pi – \cos (\pi /2)) + (\sin \pi – \sin (\pi /2)) = ( – 3(( – 1) + 3 \cdot 0)) + (0, – 1) = 2\]

2. \[\int\limits_a^b {c.\,f(x)dx = \,\,c.\,\int\limits_a^b {f(x)dx} } \]

\[\int\limits_3^8 { – 4.\ln x.dx} = – 4.\int\limits_3^8 {\ln x.dx} \]

\[\int\limits_{ – 1}^5 {5.{x^3}.dx} = 5.\int\limits_{ – 1}^5 {{x^3}.dx} \]

\[\int\limits_2^6 {\frac{{\sqrt 3 .dx}}{x}} = \sqrt 3 .\int\limits_2^6 {\frac{{dx}}{x}} \]

3. \[\int\limits_a^a {f(x)dx = 0} \]

\[\int\limits_3^3 {\ln x.dx} = 0\]

\[\int\limits_{ – 1}^{ – 1} {{x^3}.dx} = 0\]

\[\int\limits_2^2 {\frac{{dx}}{x}} = 0\]

4. \[\int\limits_a^b {f(x)dx = } \, – \int\limits_b^a {f(x)dx} \]

\[\int\limits_1^5 {3{x^2}dx = – \int\limits_5^1 {3{x^2}dx} } \]

\[3.\frac{{{x^3}}}{3}\,\,\mathop {\left. \right|}\limits_1^5 = {5^3} – {1^3} = 125 – 1 = 124\]

\[ – 3.\frac{{{x^3}}}{3}\,\,\mathop {\left. \right|}\limits_5^1 = – ({1^3} – {5^3}) = – (1 – 125) = – ( – 124) = 124\]

5. [a,c] aralığında integrallenebilir bir f fonksiyonu için, a<b<c ise;

\[\int\limits_a^c {f(x)dx = } \,\int\limits_a^b {f(x)dx} + \int\limits_b^c {f(x)dx} \]

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