Question

(sin[x])(cos[x/2])
e^arctan[x]/1+x^2

Answer 1

$$\int \sin x \cos \frac {x}{2} d x = 2 \int \sin \frac {x}{2} \cos^ {2} \frac {x}{2} d x = - 2 \cdot 2 \int \cos^ {2} \frac {x}{2} d (\cos \frac {x}{2}) = - 4 \int t ^ {2} d t = - \frac {4 t ^ {3}}{3} + C = - \frac {4}{3} \cos^ {3} \frac {x}{2} + C$$ $$\int \frac {e ^ {\arctan x}}{1 + x ^ {2}} d x = \int e ^ {\arctan x} d (\arctan x) = \int e ^ {t} d t = e ^ {t} + C = e ^ {\arctan x} + C$$