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Evaluate

$$ \int_{-\infty}^{\infty} \frac{e^{2x}}{ae^{3x}+b} dx,$$ where $a,b \gt 0$

I tried using $y=e^x$, but I still can't solve it.

I get $\displaystyle\int_0^\infty \frac y{ay^3+b} \, dy.$

Is there any different method to solve it?

jimjim
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Miras
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  • Shouldn't it be $y^2$ in the denominator? If it is, then you'll be able to use substitution. –  Jul 29 '16 at 16:18
  • @KingDuken You mean $y^2$ in the numerator ? In fact it is $y$ because the differential element $dx=dy/y$ brings a division by $y$. – Jean Marie Jul 29 '16 at 16:22
  • Notice that $ay^3 + b$ can be factored to have a linear term $(y + (b/a)^{1/3})$ and a quadratic term. Now do partial fractions. –  Jul 29 '16 at 16:26
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    @JeanMarie : It should be just as the poster wrote it. $$ \int \frac{e^{2x}}{a e^{3x} + b},dx = \int \frac{e^x}{ a\left( e^x \right)^3 + b} , \Big(e^x , dx \Big) = \int \frac y {ay^3+b} , dy\qquad\text{since } e^x,dx = dy. $$ – Michael Hardy Jul 29 '16 at 17:39

5 Answers5

5

It suffices to compute:

$$\int \frac{y}{y^3 - r^3}dy$$

(putting $r = - \sqrt[3]{\frac{b}{a} }$). Let's assume that $a, b > 0$.

After decomposing into partial fractions, i.e.:

$$\frac{y}{y^3 - r^3} = \frac1{3r} \left( \frac1{y - r} - \frac{y - r}{y^2 + ry + r^2} \right)$$

we find:

$$\int\frac{y}{y^3 -r^3} dy = \frac1{3r}\left[ \log\left( \frac{y-r}{\sqrt{y^2 + ry + r^2}} \right) + \sqrt 3 \arctan \left( \frac{2y + r}{r \sqrt 3} \right) \right]$$

Hence,

$$\int_0^{\infty} \frac{y}{y^3- r^3} dy = \frac{1}{r\sqrt 3} \left( \arctan (-\infty) - \arctan\left( \frac1{\sqrt 3} \right) \right)= -\frac{2\pi}{3r\sqrt 3}$$

and so,

$$\int_{-\infty}^{\infty} \frac{e^{3x}}{ae^{2x} + b} dx = \frac{2\pi}{3\sqrt 3 \sqrt[3]{a^2 b}}$$

  • sign needs to be changed, then it matches what I got except for a factor of $2$. – robjohn Jul 29 '16 at 17:25
  • @robjohn the sign problem was with the evaluation of the $\arctan$ integral. I fixed it. I can't see why there should be a factor of $2$ –  Jul 29 '16 at 17:32
  • Somewhere, there is a missing factor of $2$. For $a=b=1$, Mathematica gives $\frac{2\pi}{3\sqrt3}$. – robjohn Jul 29 '16 at 17:32
  • @robjohn you are right. The problem is still with the evaluation of the said integral. Now it should become correct. –  Jul 29 '16 at 17:39
  • Indeed. It now matches what I got. – robjohn Jul 29 '16 at 17:53
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$$ \begin{align} \int_{-\infty}^\infty\frac{e^{2x}}{ae^{3x}+b}\,\mathrm{d}x &=\frac1{3b}\left(\frac ba\right)^{2/3}\int_0^\infty\frac{u^{-1/3}}{u+1}\,\mathrm{d}u\tag{1}\\ &=\frac13\left(\frac1{ba^2}\right)^{1/3}\pi\csc\left(\frac23\pi\right)\tag{2}\\ &=\frac{2\pi}{3\sqrt3}\left(\frac1{ba^2}\right)^{1/3}\tag{3} \end{align} $$ Explanation:
$(1)$: $u=\frac abe^{3x}$
$(2)$: result from this answer
$(3)$: $\csc\left(\frac23\pi\right)=\frac2{\sqrt3}$

Community
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robjohn
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2

Dividing both numerator and denominator by $e^{3x}$, we get $$\frac{e^{2x}}{ae^{3x}+b} dx=\frac{e^{-x}}{a+be^{-3x}} dx$$ If you use $y=e^{-x}$, then you get $$-\frac{dy}{a+by^3}=-\frac{1}{b}\cdot\frac{dy}{y^3+r^3}$$ where $r=\sqrt[3]{\frac{b}{a}}$

Now if you substitute $y=r\tan^\frac{2}{3} \theta$ and $dy=\frac{2}{3}r\tan^{-\frac{1}{3}} \theta \sec^2 \theta \, d\theta$, your integrand will become $$-\frac{1}{b}\cdot\frac{\frac{2}{3}r\tan^{-\frac{1}{3}} \theta \sec^2 \theta \, d\theta}{r^3\sec^2 \theta}=-\frac{2}{3br^2}\cdot\tan^{-\frac{1}{3}} \theta \, d\theta$$

If you change the limits properly, then I think you will be able to complete the integration. Hope this helps.

SchrodingersCat
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WLOG, $a=b=1$. Then factoring the denominator, we decompose in simple fractions

$$\frac y{y^3+1}=A\frac1{y+1}+B\frac{2y-1}{y^2-y+1}+C\frac1{y^2-y+1}=\frac{(A+2B)y^2+(-A+B+C)y+(A-B+C)}{y^3+1}.$$

Identifying,

$$\frac y{y^3+1}=\frac13\frac1{y+1}-\frac16\frac{2y-1}{y^2-y+1}+\frac12\frac1{y^2-y+1}.$$

The first two terms have an obvious antiderivative. The third one also becomes easy by completing the square

$$\frac1{y^2-y+1}=\frac1{\left(y-\frac12\right)^2+\frac34}=\frac43\frac1{\left(\sqrt{\frac43}(y-\frac12)\right)^2+1}$$ which yields an arctangent.

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$$ ay^3 + b = a\left( y^3 + \frac b a \right) $$ $$ y^3 + \frac b a = y^3 +c^3 = (y+c)(y^2 - yc + c^2) \quad\text{where } c = \sqrt[3]\frac b a. $$ So use partial fractions to get $$ \frac \bullet {y+c} + \frac {((\bullet\, y) + \bullet)} {y^2 - yc + c^2}. $$ To integrate the second term, you have $$ \frac {ey+f}{y^2 - yc + c^2}, $$ and letting $u=y^2-yc+c^2$ so $du = (2y-c)\,dy$. Multiplying both sides of the last equality by $e/2$, we get $$ \frac e 2 \, du = \left( ey+\frac{ce}{2} \right) \, dy. $$ So $$ (ey+f)\,dy = \left( \underbrace{\left( ey + \frac{ce}2 \right)\,dy} + \left( f - \frac{ce}2 \right) \, dy \right). $$ The substitution handles the part over the $\underbrace{\text{underbrace}}.$ Then you have $$ \int \frac{dy}{y^2-yc+c^2}. $$ Complete the square $$ y^2 - yc + c^2 = \left( y^2 - yc + \frac{c^2} 4 \right) + \frac{3c^2} 4 = \left( y - \frac c 2 \right)^2 + \frac{3c^2} 4. $$ So when you integrate, you get an arctangent.

Michael Hardy
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