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$$\int_0^1\dfrac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx$$

Attempt:

If we write: $f(x)= x^4 + x^3+ x^2$, we get:

$$I = \displaystyle\int_0^1 \dfrac{3f(x)+x^3}{(f'(x)+1)^2}\, dx$$

I have no idea how to proceed. Integration by parts/ substitution can't help.

I don't need the full solution. Just a guiding hint would suffice.

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    $\begingroup$ Hint: change variable from $x \to y = \frac1x$. In the new variable, when you express the integrand as $\frac{p(y)}{q(y)^2}$, what is the relation between $p(y)$ and $q(y)$? Once you do that, the integral trivally evaluates to $\frac{1}{10}$. $\endgroup$ – achille hui Jul 5 '18 at 8:29
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    $\begingroup$ @samjoe just like last time, OP will figure that out and probably post an answer.... $\endgroup$ – achille hui Jul 5 '18 at 8:50
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    $\begingroup$ @achille I respect your opinion, but my say on this is that your comment is less of a comment but a full answer. So rather it made more sense to post it as such. Same with last question which I just saw. But is is obviously your call not mine. $\endgroup$ – samjoe Jul 5 '18 at 9:50
  • $\begingroup$ If all other “shortcut” methods don’t work, there is always the brute force method that involves factoring and then partial fraction expansion. $\endgroup$ – Colin Hicks Jul 5 '18 at 12:35
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Since this integral is very closely linked to the Quotient Rule for differentiation (due to the square in the denominator), we will try to write the numerator as an expression of the form $$P'(x)\cdot(4x^3+3x^2+2x+1)-P(x)\cdot(4x^3+3x^2+2x+1)'$$ so $$3x^4+4x^3+3x^2=P'(x)\cdot(4x^3+3x^2+2x+1)-2P(x)\cdot(6x^2+3x+1).$$

We see that $P$ must be at least a quadratic, so $P(x)=ax^2+bx+c$ for some real numbers $a,b,c$. Then $$\begin{align}3x^4+4x^3+3x^2&=(2ax+b)\cdot(4x^3+3x^2+2x+1)-2(ax^2+bx+c)(6x^2+3x+1)\\&=8ax^4+(6a+4b)x^3+(4a+3b)x^2+(2a+b)x+b-12ax^4-(6a+12b)x^3\\&\,\,\,\,\,\,-(2a+6b+12c)x^2-(2b+6c)x-2c\\&=-4ax^4-8bx^3+(2a-3b-9c)x^2+(2a-b-6c)x+b-2c\end{align}$$ Hence $$a=-\frac34,\quad b=-\frac12,\quad c=-\frac14$$ This means that $$P(x)=-\frac14(3x^2+2x+1)$$ so $$\int_0^1\dfrac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx=-\frac14\left[\frac{3x^2+2x+1}{4x^3 + 3x^2 + 2x+ 1}\right]_0^1=\frac1{10}.$$

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    $\begingroup$ Amazing! Thanks. $\endgroup$ – Abcd Jul 5 '18 at 12:38
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The hint of @achillehui to substitute $x\to y=\frac{1}{x}$ is valuable and should be an answer by its own.

We obtain \begin{align*} \color{blue}{\int_0^1\frac{3x^4+4x^3+3x^2}{\left(4x^3+3x^2+2x+1\right)^2}\,dx} &=\int_1^\infty\frac{3y^{-4}+4y^{-3}+3y^{-2}}{\left(4y^{-3}+3y^{-2}+2y^{-1}+1\right)}y^{-2}\,dy\tag{1}\\ &=\int_1^\infty\frac{3y^2+4y+3}{\left(y^3+2y^2+3y+4\right)^2}\,dy\tag{2}\\ &=-\left.\frac{1}{y^3+2y^2+3y+4}\right|_1^\infty\tag{3}\\ &\color{blue}{=\frac{1}{10}} \end{align*}

Comment:

  • In (1) we substitute $y=\frac{1}{x},\quad dy=-\frac{1}{x^2}dx$.

  • In (2) we expand with $y^6$.

  • In (3) we integrate by noting that $\frac{d}{dy}\left(y^3+2y^2+3y+4\right)=3y^2+4y+3$.

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