If an antiderivative of f(x) is e^x and an antiderivative of g(x) is x , then f(x) x , dx + g(x)e^x , dx =

Step-by-step solution

Given that an antiderivative of f(x) is e^x , we have f(x) , dx = e^x f(x) = d dx (e^x) = e^x . Given that an antiderivative of g(x) is x , we have g(x) , dx = x g(x) = d dx ( x) = - x . We need to evaluate the integral: I = f(x) x , dx + g(x)e^x , dx Substituting f(x) = e^x and g(x) = - x : I = e^x x , dx + (- x)e^x , dx I = e^x ( x - x) , dx Using the standard integral formula e^x (h(x) + h'(x)) , dx = e^x h(x) + c , where h(x) = x and h'(x) = - x : I = e^x x + c Answer: e^x x + c