∫ (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
