Tính các tích phân sau a) pi/2 ∫ 0 sin^2 x/( 1 + cos x) dx ;

8/20

Tính các tích phân sau

a) \(\int\limits_0^{\frac{\pi }{2}} {\frac{{{{\sin }^2}x}}{{1 + \cos x}}dx} \);

b) \(\int\limits_0^{\frac{\pi }{4}} {\frac{{\cos 2x}}{{\cos x\left( {1 + \tan x} \right)}}dx} \);

c) \(\int\limits_0^{\frac{\pi }{2}} {\left( {2{{\sin }^2}x + 3} \right)dx} \)

Nhập câu trả lời ở đây

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Giải thích

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

b)\(\int\limits_0^{\frac{\pi }{4}} {\frac{{\cos 2x}}{{\cos x\left( {1 + \tan x} \right)}}dx} \)\( = \int\limits_0^{\frac{\pi }{4}} {\frac{{\left( {\cos x - \sin x} \right)\left( {\cos x + \sin x} \right)}}{{\cos x + \sin x}}dx} \)\( = \int\limits_0^{\frac{\pi }{4}} {\left( {\cos x - \sin x} \right)dx} = \left. {\left( {\sin x + \cos x} \right)} \right|_0^{\frac{\pi }{4}} = \sqrt 2 - 1\).

c) \(\int\limits_0^{\frac{\pi }{2}} {\left( {2{{\sin }^2}x + 3} \right)dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {\left( {1 - \cos 2x + 3} \right)dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {\left( {4 - \cos 2x} \right)dx} \)\( = \left. {\left( {4x - \frac{1}{2}\sin 2x} \right)} \right|_0^{\frac{\pi }{2}} = 2\pi \).