4.2.10 · D1Partial fractions — linear, repeated, irreducible quadratic
Ek badi mushkil polynomial fraction asal mein kuch simple fractions ko ek common denominator par add karke bani thi — aur addition reversible hai, isliye hum use un-add karke wapas unhi simple pieces mein tod sakte hain. Neeche har ek symbol do mein se ek sawaal ka jawaab deta hai: "woh simple pieces kya the?" ya "main har simple piece ko integrate kaise karunga?"
Isse pehle ki tum partial fractions ki machine par trust karo, tumhe har woh symbol samajhna hoga jo woh silently assume karti hai. Hum unhe ek-ek karke, bilkul scratch se build karte hain, har ek ek picture ke saath anchored hai.
Ek polynomial terms ka sum hota hai, jahan har term ek number ka x x 2 − x − 2 3 x + 1 Degree woh sabse bada power hai jo appear karta hai. Toh x 2 − x − 2 2 3 x + 1 1 7 0
Degree ko "graph kitna curvy ho sakta hai" ki tarah socho: degree 1 2
Definition Rational functionEk rational function ek polynomial ko doosre polynomial se divide karna hai: Q ( x ) P ( x ) P numerator ) aur Q denominator ) dono polynomials hain. Letters P Q P particular polynomial ke liye, Q
deg deg P deg ( 3 x + 1 ) = 1 deg P < deg Q
Intuition "Proper" kyun matter karta haiFigure dekho. Jab upar ki degree neeche se choti ho (deg P < deg Q x 0 proper hai. Jab upar utna hi curvy ya zyada curvy ho (deg P ≥ deg Q improper hai, aur koi bhi sum of shrinking proper pieces kabhi iske barabar nahi ho sakta. Yahi wajah hai ki recipe pehle proper input maangti hai.
Ek polynomial ko factor karna matlab hai usse chote polynomials ke product ke roop mein likhna, jaise x 2 − x − 2 ( x − 2 ) ( x + 1 ) factor hai. (Dekho Factoring polynomials over the reals .)
Ek polynomial ka root woh value of x 0 ( x − 2 ) x = 2 x = 2 2 − 2 = 0 ( x − a ) tabhi jab x = a
Definition Linear vs. quadratic factor
Ek linear factor ki degree 1 ( x − a )
Ek quadratic factor ki degree 2 x 2 + b x + c
Definition The discriminantx 2 + b x + c discriminant yeh number hai b 2 − 4 c kya yeh quadratic real number line par zero ko hit karta hai?
b 2 − 4 c > 0 factor ho jaata hai.b 2 − 4 c = 0 b 2 − 4 c < 0 koi real roots nahi → yeh irreducible hai (real linear factors mein nahi toda ja sakta).
Worked example Discriminant in actionx 2 + 1 b = 0 , c = 1 = 0 2 − 4 ( 1 ) = − 4 < 0 x 2 − 5 x + 6 25 − 24 = 1 > 0 ( x − 2 ) ( x − 3 ) nahi hai.
Definition Repeated factor aur exponentk ( x − a ) k ( x − a ) k k exponent hai: ( x − 1 ) 2 = ( x − 1 ) ( x − 1 ) touch karta hai aur seedha cross karne ki bajaye wapas mud jaata hai.
Parent note mein repeated factors ko fractions ka poora stack diya jaata hai (x − a A 1 + ( x − a ) 2 A 2 + …
Definition Unknown numeratorsLetters A , B , C , … un numbers ke placeholders hain jo hum abhi nahi jaante . Partial fractions jawaab ko in unknowns se bhare ek template ke roop mein likhta hai, phir unhe solve karta hai. Subscripts jaise A 1 , A 2 B x + C B x C
∑ j = 1 ∑ m j 1 , 2 , … , m ∑ j = 1 3 ( x − a ) j A j = x − a A 1 + ( x − a ) 2 A 2 + ( x − a ) 3 A 3
Ek baar fraction split ho jaaye, toh har piece teen standard results mein se KISI EK se integrate hota hai. Yahan bataya gaya hai ki har tool kyun use hota hai, koi doosra nahi .
Definition Integral sign aurd x ∫ f ( x ) d x "kaun sa function f ( x ) d x x + C
Definition Substitution — glue toolIntegration by substitution hume integrand ke ek chunk ko single letter u u ∫ x 2 + 1 x d x = 2 1 ln ( x 2 + 1 ) u = x 2 + 1 x d x 2 1 d u
Definition Completing the squareCompleting the square x 2 + b x + c ( x + 2 b ) 2 + d 2 Kyun? Arctan tool ko exact shape ( stuff ) 2 + ( constant ) 2
Intuition Numerator ki degree denominator se ek kam hoti haiEk common denominator par, ek single fraction ke top ki degree uske bottom se ek neeche tak hi ja sakti hai. Ek linear bottom ( x − a ) 1 A 0 2 B x + C 1 wajah hai ki ek repeated factor ko missing freedom recover karne ke liye saare lower powers chahiye.
Discriminant tells irreducible
Khud ko test karo — har line jawaab reveal karti hai.
deg P < deg Q Numerator denominator se kam curvy hai (proper); tabhi shrinking simple pieces uske barabar sum ho sakti hain.
x 2 + b x + c Jab discriminant b 2 − 4 c < 0
x 2 − 5 x + 6 25 − 24 = 1 > 0 ( x − 2 ) ( x − 3 )
Ek quadratic factor ko numerator B x + C A Ek top apne bottom se ek degree neeche tak ja sakti hai; degree-2 1
Kaun sa integral log deta hai, aur kyun? ∫ x − a 1 d x = ln ∣ x − a ∣ ln x 1
∫ ( x − a ) − 2 d x − ( x − a ) − 1 Log sirf exponent − 1
Arctan kaam karne se pehle quadratic ko kaunsi shape mein force karna padta hai, aur kaise? ( x + 2 b ) 2 + d 2 square + const 1
arctan "Kaunse angle ki yeh tangent hai?" — yeh ek square + const 1
A 1 , A 2 Ek repeated factor ke kaunse power se har unknown belong karta hai.
