d x (x+a)^ 9 7 (x-b)^ 5 / 7 =

Step-by-step solution

Evaluate the integral I = d x (x+a)^ 9 7 (x-b)^ 5 / 7 using the substitution t = x-b x+a . Solving for x in terms of t gives x = b+ta 1-t , from which we obtain d x = a+b (1-t)^2 d t . The expressions transform as x+a = a+b 1-t and x-b = t(a+b) 1-t . Substituting into the integral yields: I = a+b (1-t)^2 d t ( a+b 1-t )^ 9 7 ( t(a+b) 1-t )^ 5 7 Simplifying the powers: I = 1 a+b t^ - 5 7 d t = 1 a+b t^ 2 7 2 7 + C = 7 2(a+b) t^ 2 7 + C Substituting back t = x-b x+a gives the final result: I = 7 2(a+b) ( x-b x+a )^ 2