If (1+x^4 ) x^3 d x=f(x) ( 1 g(x) )+ ⁻¹(h(x))+c , then h(x) [f(x)+f ( 1 x ) ]=

Step-by-step solution

aligned & Given, (1+x^4 ) x^3 d x = & f(x) ( 1 g(x) )+ ⁻¹(h(x))+c & x⁻³ (1+x^4 ) d x = & (1+x^4 ) x⁻² -2 + 1 2 ( 4 x^3 1+x^4 x⁻² ) d x = & - 1 2 x^2 (1+x^4 )+ 2 x 1+x^4 d x aligned Let x^2=t 2 x= dt =- 1 2 x^2 (1+x^4 )+ d t 1+t^2 aligned & =- 1 2 x^2 (1+x^4 )+ ⁻¹ (x^2 )+c & So f(x)= 1 2 x^2 , g(x)=1+x^4, h(x)=x^2 & =x^2 ( 1 2 x^2 + x^2 2 )= 1+x^4 2 = g(x) 2 aligned