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in Integrals calculus by (63.7k points)

Give reduction formula for ∫ cosn x dx.

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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 – I

 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.

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