Let In = ∫ cosn x dx
= ∫ cosn –1 x . cos x dx
= cosn– 1 x ∫ cos x dx + (n + 1) ∫ cosn–2 x .sinn xdx
= cosn– 1 x .sin x + (n + 1) ∫ cosn – 2 x .(1 – cos2 x)dx
= cosn– 1 x .sin x + (n + 1)In–2 – In
Thus, (1 + n + 1)In = cosn –1 x .sin x + (n + 1)In –2 + C
In = (cosn– 1 x .sin x /(n + 2)) + ((n + 1)/ (n + 2))In–2 + C
which is the required reduction formula.