Mathematics - Indefinite Integration Question with Solution | TestHub

Let $I=\int \frac{dx}{{\left(\sqrt{x}\right)}^2+{\left(\sqrt{x}\right)}^7}$

$=\int \frac{dx}{{\left(\sqrt{x}\right)}^7\left(\frac{1}{(\sqrt{x}{)}^5}+1\right)}$

Put $1+\frac{1}{{\left(\sqrt{x}\right)}^5}=t$

$\Rightarrow -\frac{5}{2}·\frac{1}{(\sqrt{x}{)}^7}dx=dt$

$\therefore I=-\frac{2}{5}\int \frac{dt}{t}=\frac{-2}{5}\log t+C$

$=\frac{-2}{5}\log \left[1+\frac{1}{{\left(\sqrt{x}\right)}^5}\right]+C$

$=-\frac{2}{5}\log \left(\frac{x^{5/2}+1}{x^{5/2}}\right)+C$

$\Rightarrow \lambda =\frac{-2}{5}$ and $a=\frac{5}{2}$

$\therefore a+\lambda =2.1>2$