Tính các tích phân sau: a) π ∫ 0 ( 2 cos x + 1 ) d x ; b) π ∫ 0 ( 1 + cot x ) s i n x d x ; c) π 4 ∫ 0 tan 2 x d x .

a)

\[\int\limits_0^\pi {\left( {2\cos x + 1} \right)dx} = \left. {\left( {2\sin x + x} \right)} \right|_0^\pi \]

                            \[ = \left( {2\sin \pi + \pi } \right) - \left( {2\sin 0 + 0} \right) = \pi .\]

b) \[\int\limits_0^\pi {\left( {1 + \cot x} \right){\rm{sinx}}dx} = \int\limits_0^\pi {\left( {1 + \frac{{\cos x}}{{{\mathop{\rm s}\nolimits} {\rm{inx}}}}} \right){\rm{sinx}}dx} \]

                                     \[ = \int\limits_0^\pi {\left( {\sin {\rm{x}} + \cos x} \right)dx} \]

                                     \[ = \left. {\left( { - \cos x + \sin {\rm{x}}} \right)} \right|_0^\pi = 2.\]

c)

\[\int\limits_0^{\frac{\pi }{4}} {{{\tan }^2}xdx} = \int\limits_0^{\frac{\pi }{4}} {\left( {\frac{1}{{{{\cos }^2}x}} - 1} \right)dx} \]

                  \[ = \left. {\left( {\tan x - x} \right)} \right|_0^{\frac{\pi }{4}}\]

                           \[ = \left( {\tan \frac{\pi }{4} - \frac{\pi }{4}} \right) - \left( {\tan 0 - 0} \right) = 1 - \frac{\pi }{4}\].