Giải bài tập 2 trang 20

Đề bài

Tính các tích phân sau:

a) $\underset{1}{\overset{2}{\int }}{x}^{4}dx$;

b) $\underset{1}{\overset{2}{\int }}\frac{1}{\sqrt{x}}dx$;

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

d) $\underset{0}{\overset{2}{\int }}{3}^{x}dx$.

Lời giải chi tiết

a) $\underset{1}{\overset{2}{\int }}{x}^{4}dx={\left(\frac{x^5}{5}\right)|}_1^2=\frac{2^5}{5}-\frac{1^5}{5}=\frac{31}{5}$.

b) $\underset{1}{\overset{2}{\int }}\frac{1}{\sqrt{x}}dx=\underset{1}{\overset{2}{\int }}{x}^{-\frac{1}{2}}dx={\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right)|}_1^2$

$=\frac{2^{\frac{1}{2}}}{\frac{1}{2}}-\frac{1^{\frac{1}{2}}}{\frac{1}{2}}=2\left(\sqrt{2}-1\right)$.

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

d) $\underset{0}{\overset{2}{\int }}{3}^{x}dx={\left(\frac{3^x}{\ln 3}\right)|}_0^2=\frac{3^2}{\ln 3}-\frac{3^0}{\ln 3}=\frac{8}{\ln 3}$.