Evaluate the integral # I = int_0^3 \ xf(x^2) \ dx #?
1 Answer
Jan 26, 2018
# 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
# 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 #
