Giải bài 64 trang 30 sách bài tập toán 12 - Cánh diều

Đề bài

Tính:

a) \(\int\limits_0^{\frac{\pi }{2}} {\sin xdx} \);

b) \(\int\limits_0^{\frac{\pi }{4}} {\cos xdx} \);

c) \(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\frac{1}{{{{\sin }^2}x}}dx} \);

d) \(\int\limits_0^{\frac{\pi }{4}} {\frac{1}{{{{\cos }^2}x}}dx} \);

e) \(\int\limits_0^{\frac{\pi }{2}} {\left( {\sin x - 2} \right)dx} \);

g) \(\int\limits_0^{\frac{\pi }{4}} {\left( {3\cos x + 2} \right)dx} \).

Lời giải chi tiết

a) \(\int\limits_0^{\frac{\pi }{2}} {\sin xdx}  = \left. { - \cos x} \right|_0^{\frac{\pi }{2}} =  - \cos \frac{\pi }{2} + \cos 0 = 1\).

b) \(\int\limits_0^{\frac{\pi }{4}} {\cos xdx}  = \left. {\sin x} \right|_0^{\frac{\pi }{4}} = \sin \frac{\pi }{4} - \sin 0 = \frac{{\sqrt 2 }}{2}\).

c) \(\int\limits_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\frac{1}{{{{\sin }^2}x}}dx}  = \left. { - \cot x} \right|_{\frac{\pi }{4}}^{\frac{\pi }{2}} =  - \cot \frac{\pi }{2} + \cot \frac{\pi }{4} = 1\).

d) \(\int\limits_0^{\frac{\pi }{4}} {\frac{1}{{{{\cos }^2}x}}dx}  = \left. {\tan x} \right|_0^{\frac{\pi }{4}} = \tan \frac{\pi }{4} - \tan 0 = 1\).

e) \(\int\limits_0^{\frac{\pi }{2}} {\left( {\sin x - 2} \right)dx}  = \left. {\left( { - \cos x - 2{\rm{x}}} \right)} \right|_0^{\frac{\pi }{2}} = \left( { - \cos \frac{\pi }{2} - 2.\frac{\pi }{2}} \right) - \left( { - \cos 0 - 2.0} \right) = 1 - \pi \).

g) \(\int\limits_0^{\frac{\pi }{4}} {\left( {3\cos x + 2} \right)dx}  = \left. {\left( {3\sin x + 2{\rm{x}}} \right)} \right|_0^{\frac{\pi }{4}} = \left( {3\sin \frac{\pi }{4} + 2.\frac{\pi }{4}} \right) - \left( {3\sin 0 + 2.0} \right) = \frac{{3\sqrt 2 }}{2} + \frac{\pi }{2}\).