Let a and b be two acute angles such that a + b = 2 and a < b . If _ a ^ b d x 1+ x = 10 , then the value of b is :

Step-by-step solution

Let I = _ a ^ b d x 1+ x . Using the property _ a ^ b f(x) d x = _ a ^ b f(a+b-x) d x and given a+b = 2 , we replace x with 2 - x : I = _ a ^ b d x 1+ ( 2 -x ) = _ a ^ b d x 1+ x I = _ a ^ b x 1+ x d x Adding the two expressions for I : 2I = _ a ^ b 1+ x 1+ x d x = _ a ^ b 1 d x = b - a Since I = 10 , we have b - a = 2 ( 10 ) = 5 . We are given b + a = 2 . Adding the two equations : (b - a) + (b + a) = 5 + 2 2b = 7 10 b = 7 20 . Answer: 7 20