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

Step-by-step solution

Given k = ₀^ 2 a x+b x (a+b)( x+ x) ,dx Multiplying both sides by (a+b) : k(a+b) = ₀^ 2 a x+b x x+ x ,dx Using the definite integral property ₀^a f(x) ,dx = ₀^a f(a-x) ,dx , replacing x with 2 -x : k(a+b) = ₀^ 2 a ( 2 -x )+b ( 2 -x ) ( 2 -x )+ ( 2 -x ) ,dx k(a+b) = ₀^ 2 a x+b x x+ x ,dx This matches the required integral. Thus, the value is k(a+b) . Answer: k(a+b)