Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Suppose the function ( ) satisfies the equation for all linear functions
MATHEMATICS · Single correct · ISC · Mathematics · Physics · Calculus
Problem
Suppose the function
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
g_n(x) = x^{2n+1} + a_n x + b_n
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
(
n
∈
N
n \in \mathbb{N}
n
∈
N
) satisfies the equation
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
\int_{-1}^{1} (px+q)g_n(x)dx = 0
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
for all linear functions
(
p
x
+
q
)
(px+q)
(
p
x
+
q
)
, then (1)
a
n
=
b
n
=
0
a_n = b_n = 0
a
n
=
b
n
=
0
(2)
b
n
=
0
;
a
n
=
−
3
2
n
+
3
b_n = 0; a_n = -\frac{3}{2n+3}
b
n
=
0
;
a
n
=
−
2
n
+
3
3
(3)
a
n
=
0
;
b
n
=
−
3
2
n
+
3
a_n = 0; b_n = -\frac{3}{2n+3}
a
n
=
0
;
b
n
=
−
2
n
+
3
3
(4)
a
n
=
3
2
n
+
3
;
b
n
=
−
3
2
n
+
3
a_n = \frac{3}{2n+3}; b_n = -\frac{3}{2n+3}
a
n
=
2
n
+
3
3
;
b
n
=
−
2
n
+
3
3
Related theory
Accumulation Concepts
Properties of Definite Integrals
Rational Functions
Polynomial Functions
Unit and temperature conversions
Second fundamental theorem of calculus
Place-value proofs with digits
Integrals of standard functions
Browse the problem library