If f(x)= ( x^2+ ^2 x 1+x^2 ) ^2 x d x and f(0)=0 , then f(1) equals

Step-by-step solution

aligned & Let f(x)= ( x^2+ ^2 x 1+x^2 ) ^2 x d x & = x^2 ^2 x+ ^2 x ^2 x 1+x^2 d x aligned aligned & = x^2 ^2 x+ ^2 x 1+x^2 d x & = x^2 (1+ ^2 x )+ ^2 x 1+x^2 d x & = x^2+ ^2 x (1+x^2 ) 1+x^2 d x & = x^2 1+x^2 d x+ ^2 x d x & = x^2+1-1 1+x^2 dx + ( ^2 x -1 dx . & = 1 d x- d x 1+x^2 + ^2 x d x- d x & =- -1 x+ ⁻¹ x+ c & Given: f(0)=0 & f(0)=- ⁻¹ 0+ ^2+c & c=0 & f(x)=- ⁻¹ x+ ^2 x aligned Now, f(1)=- ⁻¹(1)+ 1= 1- 4