If f(x) = x , then the value of e^ f(x) (x ^3 x + f(x)) 1 - (f(x))^2 dx =

Step-by-step solution

Given f(x) = x . Substituting f(x) into the integral, we get: I = e^ x (x ^3 x + x) 1 - ^2 x dx I = e^ x (x ^3 x + x) ^2 x dx I = e^ x ( x x + x ^2 x ) dx I = e^ x (x x + cosec x x) dx We know that d dx ( e^ x g(x) ) = e^ x (g'(x) - g(x) x) . Let g'(x) - g(x) x = x x + cosec x x . By observation, if we choose g(x) = -x - cosec x , then: g'(x) = -1 + cosec x x -g(x) x = (x + cosec x) x = x x + 1 Adding these gives: g'(x) - g(x) x = -1 + cosec x x + x x + 1 = x x + cosec x x . Thus, the integral evaluates to e^ x g(x