Evaluate $integral subscript 0 superscript 1 space straight x space straight e to the power of straight x squared space dx$

Solution

Let I = $integral subscript 0 superscript 1 space straight x space straight e to the power of straight x squared end exponent space dx$
Let $straight I subscript 1 space equals space integral space straight x space straight e to the power of straight x squared end exponent space dx$
Put $straight x squared space equals space straight y comma space space$ $therefore space space space 2 space straight x space dx space equals space dy$ $rightwards double arrow$ $straight x space dx space equals space 1 half dx$
$therefore$ $straight I subscript 1 space equals space 1 half integral straight e to the power of straight y dy space equals space 1 half straight e to the power of straight y space equals space 1 half straight e to the power of straight x squared end exponent space equals space straight F left parenthesis straight x right parenthesis comma space say$
$therefore$ by second fundamental theorem,
I = F(1) - F(0) = $1 half straight e to the power of 1 minus 1 half straight e to the power of 0 space equals space 1 half straight e minus 1 half space equals space 1 half left parenthesis straight e minus 1 right parenthesis$

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Integrals

Evaluate $integral subscript 0 superscript 1 space straight x space straight e to the power of straight x squared space dx$

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