Evaluate the integral, and show all steps. Could someone please help me with this integral please? Thanks!

Question

Answer 1

Recall: 1 + tan²θ = sec²θ

∫ 12x²sec⁴(2x³)tan²(2x³)dx = 2∫tan²(2x³)sec²(2x³)sec²(2x³)(6x²)dx

= 2∫tan²(2x³)(1 + tan²(2x³))sec²(2x³)(6x²)dx Let u = tan(2x³). Then du = sec²(2x³)(6x²)dx

= 2∫ u²(1 + u²)du = 2∫ (u² + u⁴)du = (2/3)u³ + (2/5)u⁵ + C = (2/3)tan³(2x³) + (2/5)tan⁵(2x³) + C