Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 36

$2θ+4ln|cos(\frac{θ}{4})|+C$

$\int(2- tan(\frac{θ}{4}) )dθ$ $=\int 2dθ$ $-\int tan(\frac{θ}{4}) dθ$ for $\int tan(\frac{θ}{4}) dθ$, let $u=\frac{θ}{4}$ $du=\frac{1}{4} dθ$ $dθ=4du$ $\int tan(\frac{θ}{4}) dθ$ $=4\int tan(u)$ $du$ $=4(-ln|cos(u)|$ $=-4ln|cos(\frac{θ}{4})|$ $\int 2dθ$ $-\int tan(\frac{θ}{4}) dθ$ $=2θ-(-4ln|cos(\frac{θ}{4})|)+C$ $=2θ+4ln|cos(\frac{θ}{4})|+C$