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$$\int \frac{\sqrt{t+2}}{e^t}\,dt$$

I have tried integration by parts, but that is leading me no where. I typed it into Wolfram Alpha, but don't know much about erf function, just know what wikipedia page stated.

Could I please have some help solving this.

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In basic terms, there is no anti derivative expressed in elementary functions –  imranfat Mar 19 at 2:51
    
@imranfat what makes an integral not expressed in elementary functions? –  yiyi Mar 19 at 3:16
    
to use your source of wolfram for example, when it refers to functions like erf, Si, Li, etc. These are all functions that are not standard, i.e. interms of sines,cosines, e-powers, polynomials, logs... Here is an article for reference: en.wikipedia.org/wiki/Nonelementary_integral –  imranfat Mar 19 at 18:51

2 Answers 2

up vote 4 down vote accepted

$$ \int \sqrt{x+2}\ e^{-x} \ dx$$

Let $t^{2} = x +2 $.

$$ = \int t e^{-(t^{2}-2)} \ 2 t \ dt = 2 e^{2} \int t \left( t \ e^{-t^{2}} \right) \ dt $$

Now integrate by parts by letting $u = t$ and $dv = t e^{-t^{2}} \ dt$.

$$ = 2 e^{2} \left( - \frac{1}{2} t e^{-t^{2}} + \frac{1}{2} \int e^{-t^{2}} \ dt \right) $$

$$ = 2e^{2} \left( - \frac{1}{2} t e^{-t^{2}} + \frac{1}{2} \frac{\sqrt{\pi}}{2} \text{erf}(t) + C \right) $$

$$ = 2e^{2} \left(- \frac{1}{2} \sqrt{x+2} \ e^{-(x+2)} + \frac{1}{2} \frac{\sqrt{\pi}}{2} \text{erf}(\sqrt{x+2}) \right) + C$$

$$ = -\sqrt{x+2} \ e^{-x} + \frac{\sqrt{\pi}}{2} e^{2} \text{erf}(\sqrt{x+2}) + C$$

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There is a natural substitution, $x^2=t+2$. Then $dt=2x\,dx$, and we end up with $$\int 2e^2 x^2 e^{-x^2}\,dx.$$ Integration by parts is now natural. We can let $u=e^2 x$ and $dv=2xe^{-x^2}\,dx$. We find that we could calculate our integral precisely if we could find an antiderivative of $e^{-x^2}$.

Unfortunately, there is no elementary function whose derivative is $e^{-x^2}$. (That fact can be proved.) There is a function whose derivative is $e^{-x^2}$. It is easy, for example, to find an infinite series that represents such a function.

However, there no finite combination of the standard functions that we see in elementary calculus books has derivative $e^{-x^2}$.

This is quite unfortunate. Let $$\Phi(x)=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-t^2/2}\,dt.$$ An antiderivative of $e^{-x^2}$ can be easily expressed in terms of $\Phi(x)$. And the function $\Phi(x)$ is extremely important. It is the cumulative distribution function of the standard normal distribution.

Because $\Phi(x)$ cannot be expressed in terms of elementary functions, there have for many years tables of $\Phi(x)$ for the important range, roughly $x=0$ to $x=3$. (These tables are of diminishing importance, since a good many pieces of software can compute it.)

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How does one compute the integral if the anti-derivative is unknown? –  yiyi Mar 19 at 3:39
1  
Numerically. You might want to scan Trapezoidal Rule, or Simpson's Rule. (The actual algorithms used by computer programs are generally more sophisticated.) –  André Nicolas Mar 19 at 3:43

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