Need help with solving an equation.

Question

If f is continuous and integral 0 to 9 f(x)dx=7, find integral 0 to 3 xf(x²)dx

Answer 1

Answer 2

f(x)dx = F(x)

F(9)-F(0) = 7

F(x) = 7x/9

7(9)/9 - 7(0)/9 = 7-0=7

f(x) = 7/9

graphically, it's a straight horizontal flat line

f(x^2) = 7/9 = f(x)

the value of x doesn't change the graph, it's still the same straight line

f(x) = f(x^2) = f(x^3) = f(x-3) = f(x^2 +27) = ....

xf(x^2) =x(7/9) = 7x/9

xf(x^2)dx = 7x^2/18 = F(x)

F(3) - F(0) = 7(3)^2/18 - 0

= 7/2 = 3.5

BUT if f(x) = 14x/81, an upward sloping straight line with slope = 14/81

then f(x)dx = F(x) = 14x^2/162 = 7x^2/81

then f(x^2) =7x^3/81

F(x) = 7x^4/324

F(3)-F(0) = 7(3)^4/324= 7(81/324) = 7/12

there's an infinite number of possible solutions, all depending on the value of f(x)

which could be linear, quadratic, cubic or higher order polynomials or a step function

but the most simple f(x) is 7/9