Integrate y = (2x + 2)^5 with respect to x

∫ (2x + 2)⁵ dx

Substitute u = 2x + 2, so that du = 2 dx. Then

∫ (2x + 2)⁵ dx = ∫ u ⁵ (du/2) = 1/2 ∫ u ⁵ du

Use the power rule:

1/2 ∫ u ⁵ du = 1/2 (1/6 u ⁶) + C = 1/12 u ⁶ + C

Replace u to get the antiderivative back in terms of x :

∫ (2x + 2)⁵ dx = 1/12 (2x + 2)⁶ + C

and the result can be simplified a bit as

(2x + 2)⁶ = 2⁶ (x + 1)⁶

→   ∫ (2x + 2)⁵ dx = 2⁴/3 (x + 1)⁶ + C = 16/3 (x + 1)⁶ + C