Hi, can someone please help me to solve and explain this question for me? 1) find the `int e^(2x) arctan e^x dx`

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tiburtius | High School Teacher | (Level 2) Educator

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First we will make substitution `t=e^x.`

`int e^(2x)arctan e^xdx=|(t=e^x),(dt=e^xdx)|=`

`int t arctan t dt`

Now we use partial integration

`=|(u=arctan t,dv=t dt),(du=1/(1+t^2),v=t^2/2)|=`

`t^2/2 arctan t- 1/2int t^2/(1+t^2)=`` `

`t^2/2 arctan t- 1/2int(1-1/(1+x^2))dt=``1/2(t^2 arctan t-t+arctan t)=`

Now we return our substitution `t=e^x.`

`1/2(e^(2x)arctan e^x-e^x+arctan e^x)` <-- Your solution

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