how to find the integral, when f is continuous ?

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Answer 1 · 2016-02-29T01:15:52

Perform a substitution and use the given integral to find that

#int_1^2t^3f(t^4)dt = 4#

Let #u = t^4 => du = 4t^3dt#

Additionally, when #t = 1# we have #u = 1# and when #t = 2# we have #u = 2^4=16#. Thus, performing the substitution,

#int_1^2t^3f(t^4)dt = 1/4int_1^2f(t^4)4t^3dt#

#=1/4int_1^16f(u)du#

#=1/4(16)#

#=4#