If ∫ ( x ) ^ 5 dx ( x ) ^ 7 + x^ 6 = λ log( x9 x^ 9 + 1 ) + C ,...

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

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

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

$⇒ - \dfrac{5}{2} \cdot \dfrac{1}{( \sqrt{x}{)}^{7}}dx = dt$

$∴ I = - \dfrac{2}{5} ∫ \dfrac{dt}{t} = \dfrac{- 2}{5}\log t + C$

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

$= - \dfrac{2}{5}\log\left(\dfrac{x^{5 / 2} + 1}{x5 / 2}\right) + C$

$⇒ \lambda = \dfrac{- 2}{5}$ and $a = \dfrac{5}{2}$

$∴ a + \lambda = 2. 1 > 2$