Question

What is `f(x) = int x^3-2x+e^x dx` if `f(1) = 3`?

Answer 1

`f(x)=x^4/4-x^2+e^x+15/4-e`

Explanation:

We will make use of the following integration rules:

`intkf(x)dx=kintf(x)dx`

`intx^ndx=(x^(n+1))/(n+1)+C`

`inte^xdx=e^x+C`

We can integrate each term separately:

`intx^3dx=(x^(3+1))/(3+1)+C=x^4/4+C`

`int-2xdx=-2intxdx=-2(x^(1+1)/(1+1))+C=(-2x^2)/2+C=-x^2+C`

`inte^xdx=e^x+C`

Thus, our function is

`f(x)=x^4/4-x^2+e^x+C`

However, we can determine `C`, the constant of integration, since we know that `f(1)=3`.

`3=1^4/4-1^2+e^1+C`

`3=1/4-1+e+C`

`3=-3/4+e+C`

`15/4-e=C`

We can plug this back into our function:

`f(x)=x^4/4-x^2+e^x+15/4-e`