What is $int_(0)^(2) xe^(x^2 + 2)dx$ ?

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Answer 1 · 2015-11-14T00:34:27

$1/2(e^6-e^2)$

which can be factored into

$e^2/2(e^2+1)(e^2-1)$

Approximately $198.02$

Perform a u-substitution

Let $u=x^2+2$

$(du)/dx=2x$

$dx=(du)/(2x)$

When $x=0$ , $u=0^2+2=2$

When $x=2$ , $u=2^2+2=6$

Now make the substitution into the integral

$int_2^6xe^u(du)/(2x)$

$1/2int_2^6e^udu$

Now integrate,

$1/2e^u$

Now evaluate

$1/2e^6-1/2e^2$

$1/2e^2(e^4-1)$

$1/2e^2(e^2+1)(e^2-1)$

This is approximately $198.02$

What is int_(0)^(2) xe^(x^2 + 2)dx int_(0)^(2) xe^(x^2 + 2)dx ?