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Q.
If $\int \frac{(\sqrt{x})^5}{(\sqrt{x})^7+x^6} d x=\alpha \ell {n}\left(\frac{x^\beta}{x^\beta+1}\right)+C$, then value of $\alpha$ and $\beta$ are respectively are
Integrals
Solution:
$ I=\int \frac{1}{(\sqrt{x})^2+(\sqrt{x})^7} d x=\int \frac{1}{(\sqrt{x})^7\left(1+\frac{1}{(\sqrt{x})^5}\right)} d x $
$\Rightarrow-\frac{5}{2} \frac{1}{(\sqrt{x})^7} d x=d t$
$ I=-\frac{2}{5} \int \frac{1}{1+t} d t=-\frac{2}{5} \ln \left(1+\frac{1}{x^{5 / 2}}\right)+C$
$=\frac{2}{5} \ln \left(\frac{x^{5 / 2}}{x^{5 / 2}+1}\right)+C$
so $\alpha=\frac{2}{5}$ and $\beta=\frac{5}{2}$