If ₀^ / 2 ( ^4 x+ ^4 x ) d x=k , then what is the value of ₀^ 20 ( ^4+ ^4 x ) d x ?

Step-by-step solution

Concept: If f ( x ) is a periodic function with period T , then ₀^ n T f(x) d x=n ₀^T f(x) d x If f(x) is periodic with period T , then f(a x) is periodic with period T |a| Calculation: aligned & Let f(x)= ( ^4 x+ ^4 x ) & f(x)= ( ^2 x )^2+ ( ^2 x )^2+2 ^2 x ^2 x-2 ^2 x ^2 x & f(x)= ( ^2 x+ ^2 x )^2-2 ^2 x ^2 x & f(x)=1- ^2 2 x 2 & f(x)=1- 1- 4 x 4 & f(x)= 3 4 + 4 x 4 aligned Since x is a function of period 2 4 x is periodic with period 2 4 = 2 f(x)= ( ^4 x+ ^4 x ) is periodic with period 2 Given, ₀^ / 2 ( ^4 x+ ^4