AP DSC School Assistant Math Model Test 2

Concept:The given integral can be solved using the standard formula $\int e^{x}\big[f(x)+f'(x)\big]\,dx = e^{x}f(x)+C$, where the integrand must be expressed in the form $e^{x}$ times a function plus its derivative.Explanation:We start with$$I=\int e^{x}\frac{1+\sin x}{1+\cos x}\,dx.$$Use the half‑angle identities:$\displaystyle 1+\cos x = 2\cos^{2}\frac{x}{2}$ and $\displaystyle \sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}$.Substitute these into the integrand:$$I=\int e^{x}\frac{1+2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^{2}\frac{x}{2}}\,dx.$$Split the fraction into two terms:$$I=\int e^{x}\left[\frac{1}{2\cos^{2}\frac{x}{2}}+\frac{2\sin\frac{x}{2}\cos\frac{x}{2}}{2\cos^{2}\frac{x}{2}}\right]dx.$$Simplify each term. Note $\frac{1}{\cos^{2}\theta}=\sec^{2}\theta$, so the first term becomes $\frac{1}{2}\sec^{2}\frac{x}{2}$. The second term simplifies to $\frac{\sin\frac{x}{2}}{\cos\frac{x}{2}}=\tan\frac{x}{2}$.Thus,$$I=\int e^{x}\left[\frac{1}{2}\sec^{2}\frac{x}{2}+\tan\frac{x}{2}\right]dx.$$Now compare with the standard form $\int e^{x}[f(x)+f'(x)]\,dx$. Choose$f(x)=\tan\frac{x}{2}$. Its derivative is $f'(x)=\frac{1}{2}\sec^{2}\frac{x}{2}$. Indeed, the integrand is exactly $e^{x}[f(x)+f'(x)]$. Therefore,$$I=e^{x}f(x)+C = e^{x}\tan\frac{x}{2}+C.$$Answer:The correct option is B: $e^{x}\tan\frac{x}{2}+c$.