Passage: Let ₀^ 2 a x+b x (a+b)( x+ x) ,dx=k . Question: What is the value of k ?

Step-by-step solution

Let I = ₀^ 2 a x+b x (a+b)( x+ x) ,dx Using the property ₀^a f(x) ,dx = ₀^a f(a-x) ,dx , we get: I = ₀^ 2 a ( 2 -x )+b ( 2 -x ) (a+b) ( ( 2 -x )+ ( 2 -x ) ) ,dx I = ₀^ 2 a x+b x (a+b)( x+ x) ,dx Adding the two equations for I , we get: 2I = ₀^ 2 a x+b x+a x+b x (a+b)( x+ x) ,dx 2I = ₀^ 2 (a+b)( x+ x) (a+b)( x+ x) ,dx 2I = ₀^ 2 1 ,dx 2I = 2 I = 4 Thus, k = 4 Answer: 4