Question

Solve this INTEGRATION!!!!

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Answer 1

Answer:

Hey mate....

Here is your answer....

Refer the attached picture.

This is be solved by integreation by partial fractions.

-Mähëshwářàñ

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\bf{Evaluate\:\::\int \dfrac{x^2}{\left(x^2+4\right)\left(x^2+9\right)}dx}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\bf{\int \frac{x^2}{\left(x^2+4\right)\left(x^2+9\right)}dx=-\frac{2}{5}\arctan \left(\frac{x}{2}\right)+\frac{3}{5}\arctan \left(\frac{x}{3}\right)+C}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\sf{\text { Take the partial fraction of } \dfrac{x^{2}}{\left(x^{2}+4\right)\left(x^{2}+9\right)}:-\dfrac{4}{5\left(x^{2}+4\right)}+\dfrac{9}{5\left(x^{2}+9\right)}}$

$=\int \:-\dfrac{4}{5\left(x^2+4\right)}+\dfrac{9}{5\left(x^2+9\right)}dx$

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=-\int \dfrac{4}{5\left(x^2+4\right)}dx+\int \dfrac{9}{5\left(x^2+9\right)}dx$

$\int \dfrac{4}{5\left(x^{2}+4\right)} d x=\dfrac{2}{5} \arctan \left(\dfrac{x}{2}\right)\\\\\\\int \dfrac{9}{5\left(x^{2}+9\right)} d x=\dfrac{3}{5} \arctan \left(\dfrac{x}{3}\right)$

$=-\dfrac{2}{5}\arctan \left(\dfrac{x}{2}\right)+\dfrac{3}{5}\arctan \left(\dfrac{x}{3}\right)$

$\tt{Add\:a\:constant\:to\:the\:solution}$

$\boxed{\bf{=-\dfrac{2}{5}\arctan \left(\dfrac{x}{2}\right)+\dfrac{3}{5}\arctan \left(\dfrac{x}{3}\right)+C}}$