Question

Find f if f′(x)=2x−3/x^4, x>0 and f(1)=2.

Answer 1

`f'(x)=2x-\frac3{x^4}`

`\displaystyle\int f'(x)\,\mathrm dx=\int\left(2x-\frac3{x^4}\right)\,\mathrm dx`

`f(x)=x^2+\frac1{x^3}+C`

Given that
`f(1)=2`, we get


`2=1^2+\frac1{1^3}+C`

`2=2+C`

`\implies C=2`

so that


`f(x)=x^2+\frac1{x^3}`