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,

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

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

$=1/4(16)$

$=4$