Giải bài 4.15 trang 13

Đề bài

Tính các tích phân sau:

a) $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(3\cos x+2\sin x\right)dx$;

b) $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{4}}{\int }}\left(\frac{1}{{\cos }^{2}x}-\frac{1}{{\sin }^{2}x}\right)dx$.

Lời giải chi tiết

a) Ta có $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(3\cos x+2\sin x\right)dx=3\underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos xdx+2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin xdx=3{\sin x|}_0^{\frac{\pi }{2}}-2{\cos x|}_0^{\frac{\pi }{2}}=3+2=5$.

b) Ta có

$\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{4}}{\int }}\left(\frac{1}{{\cos }^{2}x}-\frac{1}{{\sin }^{2}x}\right)dx={\left(\tan x+\cot x\right)|}_{\frac{\pi }{6}}^{\frac{\pi }{4}}=\tan \frac{\pi }{4}+\cot \frac{\pi }{4}-\tan \frac{\pi }{6}-\cot \frac{\pi }{6}=2-\frac{1}{\sqrt{3}}-\sqrt{3}=2-\frac{4}{\sqrt{3}}$.