If f is continuous and ? (0 to 81) f(x) dx = 14, find ? (0 to 9) xf(x^2) dx

Question

If f is continuous and

 

∫ (0 to 81) f(x) dx = 14,

find
∫ (0 to 9)  xf(x^2) dx

Answer 1

Use the method of u-substitution:

 

Let u = x².  Then du = 2xdx.  So, xdx = ½du

 

When x = 0, u = 0² = 0 and when x = 9, u = 9² = 81.

 

So, ∫xf(x²)dx from 0 to 9 = ½∫f(u)du from 0 to 81

 

                                   = ½(14) = 7