Resum conceptes bàsics del lliurament 5: Com aplicar les propietats anteriors

Com puc aplicar les propietats anteriors?

Propietat 1:

$box enclose space space integral k times f left parenthesis x right parenthesis space d x equals k times integral f left parenthesis x right parenthesis space d x space space end enclose$

Utilitat 1 : quan un nombre que està multiplicant a tota la funció ens interessa que no estigui dins la integral ja que $integral f left parenthesis x right parenthesis space d x$ la coneixem

Exemples :

$integral 3 x to the power of 5 space d x space equals 3 space times integral x to the power of 5 space d x space equals space 3 times x to the power of 6 over 6 plus C equals space 1 half x to the power of 6 plus C integral 2 cos x space d x space equals 2 times integral cos x space d x space equals space 2 space sin x space plus C integral 5 over x d x space equals integral 5 times 1 over x d x space equals 5 times integral 1 over x d x space equals 5 space ln space x space plus C integral 10 x e to the power of x squared end exponent d x space equals integral 5 times 2 x e to the power of x squared end exponent d x space equals 5 times integral 2 x e to the power of x squared end exponent d x space equals 5 e to the power of x squared space end exponent plus C$

Utilitat 2 : quan ens interessaria que hi hagi un nombre que està multiplicant a tota la funció perquè sapiguem resoldre la integral

Exemples :

$integral cos space left parenthesis 5 x right parenthesis space d x equals integral 5 over 5 cos space left parenthesis 5 x right parenthesis space d x space equals 1 fifth space sin left parenthesis 5 x right parenthesis space plus C$

En aquest cas ens interessava tenir un "5" dins de la integral ja que sabem que: $open parentheses sin left parenthesis 5 x right parenthesis close parentheses space apostrophe space equals space 5 times space cos space left parenthesis 5 x right parenthesis$

$integral fraction numerator 1 over denominator 2 x minus 6 end fraction d x equals integral 2 over 2 times fraction numerator begin display style 1 end style over denominator begin display style 2 x minus 6 end style end fraction d x equals 1 half integral 2 times fraction numerator begin display style 1 end style over denominator begin display style 2 x minus 6 end style end fraction d x equals fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction integral fraction numerator begin display style 2 end style over denominator begin display style 2 x minus 6 end style end fraction d x equals fraction numerator begin display style 1 end style over denominator begin display style 2 end style end fraction times space L n left parenthesis 2 x minus 6 right parenthesis plus C$

En aquest exemple ens interessava tenir un "2" dins de la integral ja que sabem que: $open parentheses L n left parenthesis space 2 x minus 6 right parenthesis close parentheses space apostrophe space equals space fraction numerator 2 over denominator 2 x minus 6 end fraction$

$integral x squared times e to the power of x cubed end exponent space d x space equals integral 3 over 3 times x squared times e to the power of x cubed end exponent space d x space equals 1 third integral 3 times x squared times e to the power of x cubed end exponent space d x space equals 1 third e to the power of x cubed end exponent plus C$

En aquest exemple ens interessava tenir un "3" dins de la integral ja que: $open parentheses e to the power of x cubed end exponent close parentheses to the power of space apostrophe end exponent space equals space e to the power of x cubed end exponent space times space 3 x squared$

Propietat 2 i 3:

$box enclose space space integral f left parenthesis x right parenthesis plus-or-minus g left parenthesis x right parenthesis space d x equals integral f left parenthesis x right parenthesis space d x space plus-or-minus integral g left parenthesis x right parenthesis space d x space space end enclose$

Utilitat : quan sabem resoldre la integral per separat de cada funció

Exemples :

$integral e to the power of x plus sin x space space d x equals integral e to the power of x space d x space plus integral sin x space space d x equals e to the power of x space end exponent plus left parenthesis negative cos x right parenthesis space plus C equals e to the power of x minus cos x space space plus C integral 1 over x minus 2 x e to the power of x squared end exponent space space d x equals integral 1 over x space d x space minus integral 2 x e to the power of x squared end exponent space space space d x equals ln space x space minus e to the power of x squared end exponent space space plus C$

I ara acabem amb exemples que combinen les tres propietats

Exemples :

$integral 4 e to the power of x minus 5 sin x space space d x equals 4 times integral e to the power of x space d x space minus 5 integral sin x space space d x equals 4 e to the power of x space end exponent minus 5 left parenthesis negative cos x right parenthesis space plus C equals 4 e to the power of x plus 5 cos x space space plus C integral 6 over x minus x e to the power of x squared end exponent space space d x equals integral 6 over x space d x space minus integral x e to the power of x squared end exponent d x equals 6 times integral 1 over x space d x space minus integral 2 over 2 x e to the power of x squared end exponent d x equals 6 space ln space x space minus 1 half integral 2 x e to the power of x squared end exponent d x equals space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 6 space ln space x space minus 1 half e to the power of x squared end exponent space plus space C$

Propietat intel·lectual i drets d'autoria