Find the most general form of the antiderivative of f(x)= xe^(x^2).

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Let `t = e^(x^2)`

Then `dt/dx = e^(x^2)*2x`

`dt = e^(x^2)*2xdx`

`f(x) = x*(e^(x^2))`

`intdt = int(e^(x^2)*2x)dx`

`intdt = 2*int(e^(x^2)*x)dx`

` int(e^(x^2)*x)dx` = `(1/2)*intdt`

` intf(x)dx ` = `(1/2)(t)+C`

= (1/2)(e^(x^2))+C

**So antiderivative f(x) = `(1/2)(e^(x^2))+C` where C is a constant.**

To solve the integral `int xe^{x^2}dx` consider the substitution `u=x^2`. Then `du=2xdx` which means the integral becomes

`1/2 int e^udu`

`=1/2 e^u + C`

`=1/2 e^x^2 +C`

If you have two functions multiplied together like `x` and `e^{x^2}` , where the derivative of the argument of the one function `(d/{dx}(x^2)=2x)` is the other function, then we can always make the substitution which makes the integral easier to solve.

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