How to procced in following integration

Question

We have to find the following integration .

I tried it a lot , but stuck how to proceed .

Please help

Answer 1

HINT:

Divide the numerator & the denominator by $x^{10}$ of $$F=\dfrac{5x^4+4x^5}{(x^5+x+1)^2}$$ to find $$F=\dfrac{5x^{-6}+4x^{-5}}{(1+x^{-4}+x^{-5})^2}$$

$\dfrac{d(1+x^{-4}+x^{-5})}{dx}=?$

OR

Replace $x$ with $\dfrac1y$

Answer 2

You can Directly Solved it::

Let $$I = \int\frac{4x^5+5x^4}{(x^5+x+1)^2}dx$$

Put $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt$

So $$I = -\int\frac{(4+5t)\cdot t^5}{(t^5+t^4+1)^2}\cdot \frac{1}{t^2}dt = -\int\frac{4t^3+5t^4}{(t^5+t^4+1)^2}dt$$

So we get $\displaystyle (t^5+t^4+1)=u\;,$ Then $(5t^4+4t^3)dt = du$

So $$I = -\int\frac{1}{u^2}du = \frac{1}{u}+\mathcal{C} = \frac{1}{t^5+t^4+1}+\mathcal{C}$$

So $$I = \int\frac{4x^5+5x^4}{(x^5+x+1)^2}dx = \frac{x^5}{x^5+x+1}+\mathcal{C}$$