Giải bài 64 trang 30

Tính: a) (intlimits_0^{frac{pi }{2}} {sin xdx} ); b) (intlimits_0^{frac{pi }{4}} {cos xdx} ); c) (intlimits_{frac{pi }{4}}^{frac{pi }{2}} {frac{1}{{{{sin }^2}x}}dx} ); d) (intlimits_0^{frac{pi }{4}} {frac{1}{{{{cos }^2}x}}dx} ); e) (intlimits_0^{frac{pi }{2}} {left( {sin x - 2} right)dx} ); g) (intl

Đề bài

Tính:

a) $\int_{0}^{\frac{π}{2}} \sin x d x$ ;

b) $\int_{0}^{\frac{π}{4}} \cos x d x$ ;

c) $\int_{\frac{π}{4}}^{\frac{π}{2}} \frac{1}{\sin^{2} x} d x$ ;

d) $\int_{0}^{\frac{π}{4}} \frac{1}{\cos^{2} x} d x$ ;

e) $\int_{0}^{\frac{π}{2}} (\sin x - 2) d x$ ;

g) $\int_{0}^{\frac{π}{4}} (3 \cos x + 2) d x$ .

Phương pháp giải - Xem chi tiết

Sử dụng các công thức:

• $\int \sin x d x = - \cos x + C$ .

• $\int \cos x d x = \sin x + C$ .

• $\int \frac{1}{\cos^{2} x} d x = \tan x + C$ .

• $\int \frac{1}{\sin^{2} x} d x = - \cot x + C$ .

Lời giải chi tiết

a) $\int_{0}^{\frac{π}{2}} \sin x d x = {- \cos x |}_{0}^{\frac{π}{2}} = - \cos \frac{π}{2} + \cos 0 = 1$ .

b) $\int_{0}^{\frac{π}{4}} \cos x d x = {\sin x |}_{0}^{\frac{π}{4}} = \sin \frac{π}{4} - \sin 0 = \frac{\sqrt{2}}{2}$ .

c) $\int_{\frac{π}{4}}^{\frac{π}{2}} \frac{1}{\sin^{2} x} d x = {- \cot x |}_{\frac{π}{4}}^{\frac{π}{2}} = - \cot \frac{π}{2} + \cot \frac{π}{4} = 1$ .

d) $\int_{0}^{\frac{π}{4}} \frac{1}{\cos^{2} x} d x = {\tan x |}_{0}^{\frac{π}{4}} = \tan \frac{π}{4} - \tan 0 = 1$ .

e) $\int_{0}^{\frac{π}{2}} (\sin x - 2) d x = {(- \cos x - 2 x) |}_{0}^{\frac{π}{2}}$

$= (- \cos \frac{π}{2} - 2. \frac{π}{2}) - (- \cos 0 - 2.0) = 1 - π$ .

g) $\int_{0}^{\frac{π}{4}} (3 \cos x + 2) d x = {(3 \sin x + 2 x) |}_{0}^{\frac{π}{4}}$

$= (3 \sin \frac{π}{4} + 2. \frac{π}{4}) - (3 \sin 0 + 2.0) = \frac{3 \sqrt{2}}{2} + \frac{π}{2}$ .