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$$\int \frac{x^2\;dx}{(x^2+4)^2}$$I suppose you must have to use trigonometric substitution, or something, but do not even know where to begin to solve this integral, guys, please help me!!

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    $\begingroup$ Try partial fraction first $\endgroup$ – Jean-Sébastien Sep 26 '13 at 16:40
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    $\begingroup$ Yes, you can make the reasonable substitution $x=2\tan\theta$. When the smoke clears you are integrating $\frac{1}{2}\sin^2\theta$. $\endgroup$ – André Nicolas Sep 26 '13 at 16:51
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    $\begingroup$ ..."smoke clears". My students are going to hear a lot of that from now on! $\endgroup$ – JP McCarthy Sep 26 '13 at 17:16
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Hint: $$\frac{x^2}{(x^2+4)^2}=\frac1{x^2+4}-\frac4{(x^2+4)^2}$$

$$\therefore\int\frac{x^2}{(x^2+4)^2}dx=\int\frac1{x^2+4}dx-4\int\frac1{(x^2+4)^2}dx$$

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Let \begin{align*} x &= 2\tan \theta \\ dx &= 2 \sec^2 \theta \, d \theta \\ x^2 &= 4 \tan^2 \theta \\ x^2+4 &= 4 \tan^2 \theta +4 \\ &=4 \left( \tan^2 \theta +1 \right) \\ &=4 \sec^2 \theta. \end{align*}

Then substitute,

\begin{align*} \int \frac{x^2}{\left( x^2+4 \right)^2} \, dx &=\int \frac{4 \tan^2 \theta \cdot 2 \sec^2 \theta \, d \theta}{\left( 4 \sec^2 \theta \right)^2} \\ &= \int \frac{4 \tan^2 \theta \cdot 2 \sec^2 \theta}{16 \sec^4 \theta} \, d \theta \\ &=\frac{1}{2} \int \frac{\tan^2 \theta}{\sec^2 \theta} \, d \theta \\ &= \frac{1}{2} \int \sin^2 \theta \, d \theta \\ &= \frac{1}{4} \int \left( 1- \cos (2 \theta) \right) \, d \theta \\ &= \frac{1}{4} \left( \theta - \frac{1}{2} \sin (2 \theta) \right) \\ &=\frac{1}{4}\left( \theta -\cos \theta \cdot \sin \theta \right). \end{align*}

For the back substitution,

$$x=2 \tan \theta \Rightarrow \frac{x}{2} = \tan \theta \Rightarrow \theta = \tan^{-1}\left( \frac{x}{2}\right).$$

Form a right triangle with 2 adjacent to $\theta$, and $x$ opposite $\theta$. Then the hypoteneuse is $\sqrt{x^2+4}$. We use $\theta$, and read straight from the triangle to complete the back substitution. \begin{align*} \frac{1}{4}\left( \theta -\cos \theta \cdot \sin \theta \right) &= \frac{1}{4} \left( \tan^{-1} \left( \frac{x}{2} \right) - \frac{2}{\sqrt{x^2+4}} \cdot \frac{x}{\sqrt{x^2+4}} \right) \\ &=\frac{1}{4}\left( \tan^{-1}\left(\frac{x}{2}\right) -\frac{2x}{x^2+4} \right). \end{align*}

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  • $\begingroup$ (+1): The smartest method which can avoid partial fraction. $\endgroup$ – Harry Peter Oct 8 '13 at 11:51
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Integrate by parts: $$\int\frac{x^2}{(x^2+4)^2}dx=\int x\frac{x}{(x^2+4)^2}dx=\frac{1}{2} \int x(\frac{-1}{x^2+4})'dx=\frac{1}{2}(\frac{-x}{x^2+4} +\int \frac{1}{x^2+4}dx)=\frac{1}{2}(\frac{-x}{x^2+4} +\frac{1}{2}\arctan\frac{x}{2}) +C$$

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