[TEX] \int cot^6xdx= \int [(cot^6x+1)-1]dx= \int (1-cot^2x+cot^4x)(1+cot^2x)dx-x[/TEX]
[TEX]=- \int (1-cot^2x+cot^4x)d(cotx)-x=-cotx+ \frac{1}{3}cot^3x- \frac{1}{5}cot^5x-x+C[/TEX]
[TEX]cos^3x= \frac{1}{3}(cos3x+3cosx}[/TEX]
[TEX]\Rightarrow sin^3xsin8x= \frac{1}{6}(sin11x+sin5x)+ \frac{1}{2}(sin9x+sin7x)[/TEX]
[TEX]\Rightarrow \int cos^3xsin8xdx=- \frac{1}{66}cos11x- \frac{1}{30}cos5x- \frac{1}{18}cos9x- \frac{1}[{14}cos7x+C[/TEX]
c) 1 / [ sin^2 (x) . cos^4 (x) ]
[TEX] \int \frac{1}{sin^2xcos^4x}dx= \int (cot^2x+1)(tan^2x+1)d(tanx)[/TEX]
[TEX]= \int (2+tan^2x+ \frac{1}{tan^2x})d(tanx)=2tanx+ \frac{1}{3}tan^3x- \frac{1}{tanx}+C[/TEX]
d) 1 / [ sin^3 (x) . cos^5 (x) ]
[TEX] \int \frac{1}{sin^3xcos^5x}dx= \int \frac{1}{sin^3x(1-sin^2x)^3}dsinx[/TEX]
[TEX]= \int \frac{1}{t^3(1-t^2)^3}dt (t=sinx)[/TEX]
e) 8cosx / ( 2 + căn3.sin2x - cos2x)
[TEX]2+ \sqrt{3}sin2x-cos2x=2[1-cos(2x- \frac{ \pi}{3})]=4sin^2(x- \frac{ \pi}{6})[/TEX]
Đặt [TEX]t=x- \frac{ \pi}{6} \Rightarrow cosx= \frac{1}{2}( \sqrt{3}cost-sint)[/TEX]
[TEX] \int \frac{8cosx}{2+ \sqrt{3}sin2x-cos2x}dx= \int \frac{ \sqrt{3}cost-sint}{sin^2t}dt[/TEX]
[TEX]=- \frac{ \sqrt{3}}{sint}+ \int \frac{1}{1-cos^2t}d(cost)[/TEX]
[TEX]=- \frac{ \sqrt{3}}{sint}+ \frac{1}{2} ln| \frac{1+cost}{1-cost}|+C[/TEX]