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Prove that `int_(0)^(x)e^(xt)e^(-t^(2))dt=e^(x^(2)//4)int_(0)^(x)e^(-t^(2)//4)dt`. |
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Answer» Let `I=int_(0)^(x)e^(xt)e^(-t^(2))dt` `=e^(x^(2)//4) int_(0)^(x)e^(-x^(2)//4)e^(xt)e^(-t^(2))dt` `=e^(x^(2)//4)int_(0)^(x)e^(-(x^(2)//4-tx+t^(2)))dt` `=e^(x^(2)//4)int_(0)^(x)e^(-x//2-t^(2))dt` The result clearly suggests that we have to substitute `y//2` for `x//2-t`. Then `dt=-dy//2`. Also when `t=0,y=x`, and when `t=x, y=-x`. Thus, `I=e^(x^(2)/4)int_(x)^(-x)e^(-y^(2)//4)(-dy//2)` `=(e^(x^(2)//4))/2int_(-x)^(x)e^(-y^(2)//4)dy` `(e^(x^(2)//4))/2 2int_(0)^(x)e^(-y^(2)//4)dy` [`e^(y^(2)//4)` is an even function] `=e^(x^(2)//4)int_(0)^(x)e^(-t^(2)//4)dt` |
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