If e^x e ^ 2 x +4 e ^x+13 d x= |e^ a x +2+ e ^ 2 x +4 e ^x+13 |+ c , (where c is the constant of integration), then the value of a is equal to

Step-by-step solution

Evaluate the integral I = e^x e^ 2x +4e^x+13 ,dx for which the solution is asserted to equal |e^ ax +2+ e^ 2x +4e^x+13 |+c , and determine constant a . Substitute with t = e^x , hence dt = e^x ,dx . The integral simplifies to I = dt t^2+4t+13 . The quadratic expression t^2+4t+13 is restructured by completing the square to (t+2)²+9 = (t+2)²+3² , so I = dt (t+2)²+3² . Using the standard form dx x²+a² = |x+ x²+a² |+C , where x = t+2 and a=3 , integrate to obtain I = |(t+2)+ (t+2)²+3² |+C . Substitute back t = e^x to y