2 Answers

Kalyan Pvs · Answer 1 · 2024-02-08T15:29:36+0000

Explanation:

Hello, We are required to use the FTOC pt.1 Before we do this , know that we can break up the integral to have 0 as a limit of integrand.

$int(3x,x {}^(3) )(e {}^(t) \tan(t) )dt$

Using properties of integrals, this becomes

$int(0, {x}^(3) )(e {}^(t) \tan(t) )dt - int(0,3x)( {e}^(t) \tan(t) )dt$

Using the chain rule, the integral of the first one computes to

$e { }^{ {x}^(3) } \tan( {x}^(3) 3 {x}^(2) )$

The integral of the second one computes to

$3e {}^(3x \tan(3x) )$

So we end up getting

$(dy)/(dx) = {e}^{ {x}^(3) } \tan( {x}^(3) ) 3 {x}^(2) - 3e {}^(3x) \tan(3x)$

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