Question

Tính các tích phân sau:

a) \(\int\limits^2_1x^4dx;\)                   b) \(\int\limits^2_1\dfrac{1}{\sqrt{x}}dx;\)                 c) \(\int\limits^{\dfrac{\pi}{4}}_0\dfrac{1}{\cos^2x}dx;\)               d) \(\int\limits^2_03^xdx.\)

Answer 1

a) \(\int\limits_1^2 {{x^4}dx}  = \left. {\left( {\frac{{{x^5}}}{5}} \right)} \right|_1^2 = \frac{{{2^5}}}{5} - \frac{{{1^5}}}{5} = \frac{{31}}{5}\)

b) \(\int\limits_1^2 {\frac{1}{{\sqrt x }}dx}  = \int\limits_1^2 {{x^{ - \frac{1}{2}}}dx}  = \left. {\left( {\frac{{{x^{\frac{1}{2}}}}}{{\frac{1}{2}}}} \right)} \right|_1^2 = \frac{{{2^{\frac{1}{2}}}}}{{\frac{1}{2}}} - \frac{{{1^{\frac{1}{2}}}}}{{\frac{1}{2}}} = 2\left( {\sqrt 2  - 1} \right)\)

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

d) \(\int\limits_0^2 {{3^x}dx}  = \left. {\left( {\frac{{{3^x}}}{{\ln 3}}} \right)} \right|_0^2 = \frac{{{3^2}}}{{\ln 3}} - \frac{{{3^0}}}{{\ln 3}} = \frac{8}{{\ln 3}}\)