What is f(x) = int x^3-2x+e^x dx if f(1) = 3 ?
1 Answer
Feb 14, 2016
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
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