`f''(x) = x^6 - 4x^4 + x + 1` Find `f`.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`f''(x)=x^6-4x^4+x+1`

`f'(x)=intf''(x)dx`

`f'(x)=int(x^6-4x^4+x+1)dx`

`f'(x)=x^7/7-4(x^5/5)+x^2/2+x+c_1`

`f'(x)=x^7/7-(4x^5)/5+x^2/2+x+c_1`

`f(x)=intf'(x)dx`

`f(x)=int(x^7/7-(4x^5)/5+x^2/2+x+c_1)dx`

`f(x)=1/7(x^8/8)-4/5(x^6/6)+1/2(x^3/3)+x^2/2+c_1x+c_2`

c_1 and c_2 are constants , simplifying the above yields,

`f(x)=x^8/56-2/15x^6+x^3/6+x^2/2+c_1x+c_2`