Mathematics

Evaluate : 
$$\displaystyle\int {\dfrac{{{x^5}}}{{{x^2} + 9}}}dx $$


SOLUTION

Consider the given integral.


$$I=\displaystyle\int{\dfrac{{{x}^{5}}}{{{x}^{2}}+4}}dx$$


 


Let $$t={{x}^{2}}+4$$


$$ \dfrac{dt}{dx}=2x+0 $$


$$ \dfrac{dt}{2}=xdx $$


 


Therefore,


$$ I=\dfrac{1}{2}\displaystyle\int{\dfrac{{{\left( t-4 \right)}^{2}}}{t}}dt $$


$$ I=\dfrac{1}{2}\displaystyle\int{\dfrac{{{t}^{2}}+16-8t}{t}}dt $$


$$ I=\dfrac{1}{2}\displaystyle\int{\left( t+\dfrac{16}{t}-8 \right)}dt $$


$$ I=\dfrac{1}{2}\left[ \dfrac{{{t}^{2}}}{2}+16\log \left( t \right)-8t \right]+C $$


 


On putting the value of $$t$$, we get


$$I=\dfrac{1}{2}\left[ \dfrac{{{\left( {{x}^{2}}+4 \right)}^{2}}}{2}+16\log \left( {{x}^{2}}+4 \right)-8\left( {{x}^{2}}+4 \right) \right]+C$$


 


Hence, this is the answer.

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