say I have to integrate the following: $e^{(x^2 + bx )}$ w.r.t x
How can it be done.. I have tried taking $x^2+bx = t$ but I am stuck at $(2x + b)dx = dt$
How to integrate exponential of complicated expressions
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$\begingroup$
integration
substitution
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3the solution containes the Error-function – 2017-02-08
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0Not all functions have closed-form antiderivatives. There is no function $\int e^{x^2 + bx}dx$ that can be expressed in terms of the usual functions. – 2017-02-08
1 Answers
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$t=x+\frac{b}{2}\\ dt=dx\\ t^2=x^2+bx+\frac{b^2}{4}\\ x^2+bx = t^2 - \frac{b^2}{4}$
We have then: $\int e^{x^2+bx}dx = e^{-\frac{b^2}{4}}\int e^{t^2} dt$
But $\int e^{t^2} dt$ is non-elementary. Therefore this integral can't be expressed in terms of elementary functions.