This question can be done by using change of variable method

integral is

∫ x⁸ (x⁹ - 10)⁵ dx

here, the best choice of variable is,

let

u = x⁹ - 10

du = 9 x⁸ dx

x⁸ dx = du/9 ( using substitution )

this integral can be written as : ∫ x⁸ (x⁹ - 10)⁵ dx = ∫ (( u )⁵ / 9 )du

(1/9) ∫ ( u )⁵ du

now we know that ∫ xⁿ = xⁿ⁺¹ / n+1

= (1/9) * u⁶ / 6

=u⁶ / 54

= ( x⁹ - 10 )⁶ / 54

the answer of this indefinite integral will be ( x⁹ - 10 )⁶ / 54