Tìm nguyên hàm của hàm số sau: ∫dx/sin3x.cos5x

Question

\(\int \frac{{\rm d}x}{\sin ^{3} x.{\rm cos}^{5} x}\)

Answer 1

\(I_{3} =\int \frac{{\rm d}x}{\sin ^{3} x.{\rm cos}^{5} x} =\int \frac{{\rm d}x}{\frac{\sin ^{3} x}{{\rm cos}^{3} x} .{\rm cos}^{8} x}\) 

\(=\int \frac{{\rm d}x}{\tan ^{3} x.{\rm cos}^{6} x.{\rm cos}^{2} x}\)
Đặt  \(t=\tan x,\, \, \, {\rm d}t=\frac{1}{{\rm cos}^{2} x} {\rm d}x\), ta có
\(I_{3} =\int \frac{\left(1+t^{2} \right)^{3} }{t^{3} } {\rm d}t =\int \frac{t^{6} +3t^{4} +3t^{2} +1}{t^{3} } {\rm d}t\)
\(=\frac{t^{4} }{4} +\frac{3t^{2} }{2} +3\ln \left|t\right|-\frac{1}{2t^{2} } +C\)
Với \(t=\tan x\), ta có
\(I_{3} =\frac{\tan ^{4} x}{4} +\frac{3\tan ^{2} x}{2} +3\ln \left|\tan x\right|-\frac{1}{2\tan ^{2} x} +C\)