Question

Evaluate the integral `I = int_0^3 \ xf(x^2) \ dx`?

Answer 1

`int_0^3 \ xf(x^2) \ dx = 3/2`

Explanation:

We seek:

`I = int_0^3 \ xf(x^2) \ dx`

We can perform a substitution:

Let `u=x^2 => (du)/dx = 2x`

And the substitution will require a change in limits:

When `x={ (0),(3) :} => u={ (0),(9) :}`

Substituting into the integral, changing the variable of integration from `x` to `u` we get:

`I = 1/2 \ int_0^3 \ 2xf(x^2) \ dx`

`\ \ = 1/2 \ int_0^9 \ f(u) \ du`

`\ \ = 1/2 \ (3)`, as `int_0^9 \ f(u) \ du = int_0^9 \ f(x) \ dx`

`\ \ = 3/2`