What is f(x) = int xsinx^2 + sec^2x dx if f(0)=-1 ?
1 Answer
Sep 25, 2016
Explanation:
f(x)=intxsin(x^2)dx+intsec^2(x)dx
The second integral is simple: since
f(x)=intxsin(x^2)dx+tan(x)+C
For the remaining integral, use the substitution
f(x)=1/2int2xsin(x^2)dx+tan(x)+C
f(x)=1/2intsin(u)du+tan(x)+C
Since
f(x)=-1/2cos(u)+tan(x)+C
f(x)=-1/2cos(x^2)+tan(x)+C
Using the initial condition
-1=-1/2cos(0)+tan(0)+C
-1=-1/2(1)+0+C
C=-1/2
Thus:
f(x)=-1/2cos(x^2)+tan(x)-1/2