Chapter 1.7, Problem 419E

In the following exercises, solve for the antiderivative ${\displaystyle \int f}$ of with C = 0, the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antidelivative gives the same value as the definite integral $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ .

419. [T] ${\displaystyle \int \frac{\left(\sin x+x\cos x\right)}{1+x^2{\sin }^{2}x}}dx$ over [−6, 6]