Tìm (I = int {left( {x – 2sin x} right)frac{{dx}}{{{rm{co}}{{rm{s}}^2}x}}} )

Ta có (begin{array}{l}
I = int {left( {x – 2sin x} right)frac{{dx}}{{{rm{co}}{{rm{s}}^2}x}}}  = int {xfrac{{dx}}{{{rm{co}}{{rm{s}}^2}x}}}  – int {2sin xfrac{1}{{{rm{co}}{{rm{s}}^2}x}}} dx= {I_1} – {I_2}
end{array})

Xét ({I_1} = int {xfrac{{dx}}{{{rm{co}}{{rm{s}}^2}x}}} )

Đặt (left{ begin{array}{l}
u = x\
dv = frac{{dx}}{{{{cos }^2}x}}
end{array} right. Rightarrow left{ begin{array}{l}
du = dx\
v = tan x
end{array} right.)

(begin{array}{l}
{I_1} = xtan x – int {tan xdx = xtan x – int {frac{{sin x}}{{cos x}}} dx} \
 = xtan x – int {frac{{dleft( {sin x} right)}}{{cos x}} = } xtan x + ln cos x + C
end{array})

){I_2} = int {2sin xfrac{1}{{{{cos }^2}x}}} dx = int {frac{{ – 2dleft( {cos x} right)}}{{{{cos }^2}x}}}  = frac{2}{{cos x}} + C)