If 3 x x 4 x+7 ~d x= A x- B (4 x+7)+ c where c is the constant of integration, then the value of A+B is equal to

Step-by-step solution

The integral is evaluated using substitution. Let u = x , so du = x , dx . Substituting transforms the integral to I = 3u 4u+7 , du . After rewriting the integrand: 3u 4u+7 = 3 4 ( 1 - 7 4u+7 ) Integration yields: I = 3 4 u - 21 4 1 4u+7 , du Evaluating the remaining integral: 1 4u+7 , du = 1 4 |4u+7| Substituting back: I = 3 4 u - 21 16 |4u+7| + c Returning to the original variable u = x , and noting 4 x + 7 > 0 for all real x , gives: I = 3 4 x - 21 16 (4 x + 7) + c From the form A x - B (4 x + 7) + c , we identi