Tusaale Su'aalaha Doodda Isku-dhafan ee aan la cayimin
Isku-dhafka aan xadidnayn waa fikrad aasaasi ah oo ku jirta kalkulus, oo loo isticmaalo in lagu helo shaqada asalka ah ee laga soo qaatay shaqo la soo saaray. Isku-dhafka aan xadidnayn waxaa lagu tilmaamayaa calaamadda ∫ oo ay ku xigto shaqada la isku darayo iyo isbeddelka isku-dhafka. Maqaalkan, waxaan ka wada hadli doonnaa dhowr tusaale oo isku-dhafka aan xadidnayn iyo xalalkooda.
Su'aal Tusaale ah 1: Isku-dhafka Shaqooyinka Polynomial
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = 3x^2 \).
Dood: Si loo mideeyo shaqooyinka polynomial, waxaan isticmaalnaa xeerarka aasaasiga ah ee isdhexgalka, kuwaas oo kala ah:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Iyadoo la adeegsanayo xeerarkan, isku-dhafka \( 3x^2 \) waa:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
Markaa, \( \int 3x^2 \, dx = x^3 + C \).
Su'aal Tusaale 2: Isku-dhafka Hawlaha Jihaynta
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = e^x \).
Dood: Isku-dhafka shaqada dheeraadka ah \( e^x \) waa mid aad u fudud sababtoo ah shaqada \( e^x \) waa shaqo aan isbeddelin marka loo eego hawlgallada kala duwan iyo kuwa isku-dhafan labadaba:
\[ \int e^x \, dx = e^x + C \]
Markaa, \( \int e^x \, dx = e^x + C \).
Su'aal Tusaale ah 3: Isku-dhafka Shaqooyinka Trigonometric
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = \sin(x) \).
Dood: Si aan u mideyno hawlaha trigonometric, waxaan u baahanahay inaan ogaano isku-dhafka aasaasiga ah ee hawlahaas. Mid ka mid ah xiriirka aasaasiga ah waa:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Markaa, \( \int \sin(x) \, dx = -\cos(x) + C \).
Su'aal Tusaale ah 4: Isku-dhafka Shaqooyinka Jajabyada
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = \frac{1}{x} \).
Dood: Isku-dhafka shaqada \( \frac{1}{x} \) waa:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Markaa, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Su'aal Tusaale ah 5: Isku-dhafka shaqooyinka jibaaran ee taban
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = x^{-2} \).
Dood: Wixii \( n \neq -1 \), waxaan u isticmaalnaa xeerka aasaasiga ah ee isku-dhafka ah:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Xaaladdan, \( n = -2 \), sidaas darteed:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Markaa, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Su'aal Tusaale ah 6: Isku-dhafka Hawlaha Isku-dhafka ah
Su'aal: Go'aami isku-dhafka shaqada \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Dood: Waxaan erey kasta si gaar ah isugu dari karnaa annagoo adeegsanayna xeerarka aasaasiga ah ee isdhexgalka:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Hadda waxaan si gaar ah u dhexgelinnaa erey kasta:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Marka la isku daro natiijooyinkan, waxaan helnaa:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Markaa, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Gabagabo
Isku-dhafka aan xadidnayn waa fikrad aad muhiim u ah xisaabinta waxayna leedahay xeerar kala duwan oo sahlaya in la isku daro noocyada kala duwan ee hawlaha. Maqaalkan, waxaan ka wada hadalnay dhowr tusaale oo isku-dhafan oo aan xadidnayn, oo ay ku jiraan polynomials, exponentials, trigonometric functions, fractions, functions with negative exponents, and combination of functions. Fahmidda iyo barashada xeerarkan aasaasiga ah ee isku-dhafan waxay aad waxtar ugu yeelan doontaa xallinta dhibaatooyinka kala duwan ee xisaabinta.
Isku-dhafka aan xadidnayn ma aha oo kaliya muhiim u ah aragtida xisaabta, laakiin sidoo kale waxay leeyihiin codsiyo ballaaran oo ku saabsan fiisigiska, injineernimada, iyo meelaha kale. Iyadoo la adeegsanayo ku filan, isku-darka hawlaha kala duwan waxay noqon doontaa mid fudud oo dareen badan leh.
