Let I= x^2 (x x+ x)^2 d x . Which of the following options is equivalent to the given indefinite integral (ignoring arbitrary constant)?

Step-by-step solution

I= x^2 (x x+ x)^2 d x Put x^2=x x x xI= x x x x (x x+ x)^2 d x Integrating by parts taking x x as first function and x x (x x+ x)^2 as second function, we get I=x x x x (x x+ x)^2 d x+ (x x x+ x) ( x x (x x+ x)^2 d x ) d x ...(1) I₁= x x (x x+ x)^2 d x Put x x+ x=t x x d x=d t I₁= 1 t^2 d t I₁= -1 t (Ignoring arbitrary constant as per the question) I₁= -1 x x+ x Putting value of I₁ in (1), we get I= -x x x x+ x + x(1+ x) x x+ x d x which is the option d. Simplifying this further by taking x common in denominator in