(x-1) (x+1)^3 e^x d x=P(x)+c then P(x)=

Step-by-step solution

The integral is given by I = x-1 (x+1)^3 e^x dx . Rewrite the numerator x-1 as (x+1) - 2 : I = (x+1) - 2 (x+1)^3 e^x dx = ( x+1 (x+1)^3 - 2 (x+1)^3 ) e^x dx I = ( 1 (x+1)^2 - 2 (x+1)^3 ) e^x dx Recall the standard integral form e^x [f(x) + f'(x)] dx = e^x f(x) + c . Let f(x) = 1 (x+1)^2 . Then f'(x) = -2(x+1)⁻³ = - 2 (x+1)^3 . Substituting these into the integral formula: I = e^x ( 1 (x+1)^2 ) + c = e^x (x+1)^2 + c . Thus, P(x) = e^x (x+1)^2 . Answer: e^x (x+1)^2