What is f(x) = int xe^(x^2-1)+2x dx if f(0) = -4 ?
1 Answer
Explanation:
Split up the integral:
f(x)=intxe^(x^2-1)dx+int2xdx
For the first integral, use substitution: let
Multiply the integrand by
f(x)=1/2int2xe^(x^2-1)dx+int2xdx
Substituting in
f(x)=1/2inte^udu+2intxdx
Note that
f(x)=1/2e^u+2intxdx
Since
f(x)=1/2e^(x^2-1)+2intxdx
In order to integrate
Applying this rule:
f(x)=1/2e^(x^2-1)+2((x^(1+1))/(1+1))+C
f(x)=1/2e^(x^2-1)+x^2+C
Now, we can determine the value of
-4=1/2e^(0^2-1)+0^2+C
-4=1/2e^-1+C
-4-1/(2e)=C
Hence:
f(x)=1/2e^(x^2-1)+x^2-4-1/(2e)