Giải bài 64 trang 30

Đề bài

Tính:

a) $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin xdx$;

b) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\cos xdx$;

c) $\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{{\sin }^{2}x}dx$;

d) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{1}{{\cos }^{2}x}dx$;

e) $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\sin x-2\right)dx$;

g) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\left(3\cos x+2\right)dx$.

Lời giải chi tiết

a) $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin xdx={-\cos x|}_0^{\frac{\pi }{2}}=-\cos \frac{\pi }{2}+\cos 0=1$.

b) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\cos xdx={\sin x|}_0^{\frac{\pi }{4}}=\sin \frac{\pi }{4}-\sin 0=\frac{\sqrt{2}}{2}$.

c) $\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{{\sin }^{2}x}dx={-\cot x|}_{\frac{\pi }{4}}^{\frac{\pi }{2}}=-\cot \frac{\pi }{2}+\cot \frac{\pi }{4}=1$.

d) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{1}{{\cos }^{2}x}dx={\tan x|}_0^{\frac{\pi }{4}}=\tan \frac{\pi }{4}-\tan 0=1$.

e) $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\sin x-2\right)dx={\left(-\cos x-2x\right)|}_0^{\frac{\pi }{2}}$

$=\left(-\cos \frac{\pi }{2}-2.\frac{\pi }{2}\right)-\left(-\cos 0-2.0\right)=1-\pi$.

g) $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\left(3\cos x+2\right)dx={\left(3\sin x+2x\right)|}_0^{\frac{\pi }{4}}$

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