The value of ∫ 0 π ( ∑ r = 0 3 a r cos 3 − r x sin r x ) d x depends upon

Step-by-step solution

Let, I = ∫ 0 π ∑ r = 0 3 a r cos 3 - r ⁡ x sin r ⁡ x d x = ∫ 0 π a 0 cos 3 ⁡ x d x + ∫ 0 π a 1 cos 2 ⁡ x sin ⁡ x d x + ∫ 0 π a 2 cos ⁡ x sin 2 ⁡ x d x + ∫ 0 π a 3 sin 3 ⁡ x d x Since, ∫ 0 2 a f x d x = 2 ∫ 0 a f x d x , i f f 2 a - x = f ( x ) 0 , i f f 2 a - x = - f ( x ) ∴ Integral Ist and IIIrd become zero. ∴ The given integral depends upon a 1 and a 3 .