Tìm: a) ∫ cos 2 x 1 − s i n x d x ; b) ∫ ( 1 + 3 sin 2 x 2 ) d x ; c) ∫ 2 cos 3 x + 3 cos 2 x d x .

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Tìm:

a) \[\int {\frac{{{{\cos }^2}x}}{{1 - {\mathop{\rm s}\nolimits} {\rm{inx}}}}dx} \];

b) \[\int {\left( {1 + 3{{\sin }^2}\frac{x}{2}} \right)dx} \];

c) \[\int {\frac{{2{{\cos }^3}x + 3}}{{{{\cos }^2}x}}dx} \].

Nhập câu trả lời ở đây

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Giải thích

a)\[\int {\frac{{{{\cos }^2}x}}{{1 - \sin {\rm{x}}}}dx} = \int {\frac{{1 - {{\sin }^2}x}}{{1 - \sin {\rm{x}}}}dx} \]

\[ = \int {\frac{{\left( {1 - \sin {\rm{x}}} \right)\left( {1 + \sin {\rm{x}}} \right)}}{{\left( {1 - \sin {\rm{x}}} \right)}}dx} \]

\[ = \int {\left( {1 + \sin {\rm{x}}} \right)dx} = x - \cos x + C.\]

b) \[\int {\left( {1 + 3{{\sin }^2}\frac{x}{2}} \right)dx} = \int {\left( {1 + 3.\frac{{1 - \cos x}}{2}} \right)dx} \]

\[ = \int {\left( {\frac{5}{2} - \frac{3}{2}\cos x} \right)dx} \]

\[ = \frac{5}{2}x - \frac{3}{2}\sin {\rm{x}} + C\].

c) \[\int {\frac{{2{{\cos }^3}x + 3}}{{{{\cos }^2}x}}dx} = \int {\left( {2\cos x + \frac{3}{{{{\cos }^2}x}}} \right)dx} \]

\[ = \int {2\cos xdx + \int {\frac{3}{{{{\cos }^2}x}}dx} } \]

\[ = 2\sin x + 3\tan x + C\]