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Evaluate the indefinite integral integrate of x^3*5^(-x^2)dx

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You should use the following substitution such that:

`x^2 = t => 2xdx = dt => xdx = (dt)/2`

`int x^2*5^(-x^2)*xdx = int t*5^(-t) (dt)/2`

You need to use integration by parts such that:

`int udv = uv - int vdu`

`u = t => du = dt`

`dv = 5^(-t) dt => v = -5^(-t)/(ln 5)`

`int t*5^(-t) dt = -t*5^(-t)/(ln 5)+ int 5^(-t)/(ln 5)`

`int t*5^(-t) dt = -t*5^(-t)/(ln 5) - 5^(-t)/(ln^2 5) + c`

Substituting back `x^2`  for t yields:

`int x^2*5^(-x^2)*xdx = (1/2)(-x^2*5^(-x^2)/(ln 5) - 5^(-x^2)/(ln^2 5)) + c`

Hence, evaluating the given integral yields `int x^3*5^(-x^2)dx = (1/2)(-x^2*5^(-x^2)/(ln 5) - 5^(-x^2)/(ln^2 5)) + c.`

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