To calculate ∫ sin^2 θ sin6θ dθ, we must use trigonometric identities, such as the double-angle formula and power-reducing formulas, to simplify the integral before finding its solution.
- To calculate the integral ∫ sin^2 θ sin6θ dθ, we need to apply trigonometric identities to simplify the expression before integrating.
- One useful identity is the double-angle formula for sine, which states that sin2θ = 2sinθ cosθ.
- This can be utilized to break down the original integral into a form that is more manageable.
- However, the integral provided lacks a straightforward application of basic trigonometric identities or double-angle formulas.
- A more advanced technique involves using power-reducing formulas to express sin^2 θ and sin6θ in terms of cosines of multiple angles and then applying the sum-to-product identities.
- After which, the integral becomes a sum of simpler integrals that can be easily evaluated.