The area bounded by y=f(x), x= 1 2 , x= 3 2 and the x -axis is A sq. units, where f(x)=x+ 2 3 x^3+ 2 3 4 5 x^5+ 2 3 4 5 6 7 x^7+ and |x| < 1 . If A= ^a b , where a, b N , then find the value of (a+b) .

Step-by-step solution

aligned f(x) & =x+ 2 3 x^3+ 2 3 4 5 x^5+ 2 3 4 5 6 7 x^7+ f^ (x) & =1+x (2 x+ 2 3 4 x^3+ 2 3 4 5 6 x^5+ ) & =1+x d d x (x f(x))=1+x f(x)+x^2 f^ (x) aligned (1-x^2 ) f^ (x)=1+x f(x) , which is a linear differential equation. Also, f(0)=0 f(x)= ⁻¹ x 1-x^2 Hence, Area (A)= _ 1 / 2 ^ 3 / 2 ⁻¹ x 1-x^2 d x= ^2 24 a+b=2+24=26