Let-P-t-denote-a-given-cubic-polynomial-Find-the-constants-a-1-u-1-a-2-and-u-2-such-that-1-1-P-t-dt-a-1-P-u-1-a-2-P-u-2-

Question Number 1778 by 112358 last updated on 23/Sep/15

L e t P (t) d e n o t e a g i v e n c u b i c

p o l y n o m i a l . F i n d t h e c o n s t a n t s

a_{1}, u_{1}, a_{2} a n d u_{2} s u c h t h a t

\int_{- 1}^{1} P (t) d t = a_{1} P (u_{1}) + a_{2} P (u_{2}) .

Answered by Rasheed Soomro last updated on 23/Sep/15

L e t P (t) = {a t}^{3} + {b t}^{2} + c t + d

L H S :

\int P (t) d t = \frac{a}{4} t^{4} + \frac{b}{3} t^{3} + \frac{c}{2} t^{2} + d t + C

\int_{- 1}^{1} P (t) d t = {[\frac{a}{4} t^{4} + \frac{b}{3} t^{3} + \frac{c}{2} t^{2} + d t + C]}_{- 1}^{1}

= [\frac{a}{4} + \frac{b}{3} + \frac{c}{2} + d + C] - [\frac{a}{4} - \frac{b}{3} + \frac{c}{2} - d + C]

= \frac{2 b}{3} + 2 d

R H S :

a_{1} P (u_{1}) + a_{2} P (u_{2})

= a_{1} {a {(u_{1})}^{3} + b {(u_{1})}^{2} + c (u_{1}) + d} + a_{2} {a {(u_{2})}^{3} + b {(u_{2})}^{2} + {c u}_{2} + d}

C o m p a r i n g c o e f f i c i e n t s o f b a n d d i n L H S a n d R H S

\frac{2}{3} = a_{1} {(u_{1})}^{2} + a_{2} {(u_{2})}^{2} a n d 2 = a_{1} + a_{2}

S o l v i n g s i m u l t a n e o u s l y

3 {(u_{1})}^{2} a_{1} + 3 {(u_{1})}^{2} a_{2} = 6 {(u_{1})}^{2} \dots \dots \dots \dots I

3 {(u_{1})}^{2} a_{1} + 3 {(u_{2})}^{2} a_{2} = 2 \dots \dots \dots \dots I I

S u b t r a c t i n g I I f r o m I

3 {{(u_{1})}^{2} - {(u_{2})}^{2}} a_{2} = 6 {(u_{1})}^{2} - 2

a_{2} = \frac{6 {(u_{1})}^{2} - 2}{3 {{(u_{1})}^{2} - {(u_{2})}^{2}}}

S i m i l a r l y, a_{1} = \frac{6 {(u_{2})}^{2} - 2}{3 {{(u_{2})}^{2} - {(u_{1})}^{2}}}

I t h i n k u_{1} a n d u_{2} s h o u l d b e t a k e n a s a r b i t r a r y c o n s t a n t s .

O r o t h e r w i s e i f a_{1} a n d a_{2} a r e t a k e n a s a r b i t r a r y t h e n

u_{1} a n d u_{2} c a n b e d e t e r m i n e d .