Question

Tìm:

a) \(\int\dfrac{1}{x^4}dx;\)                     b) \(\int x\sqrt{x}dx\) (x > 0);                  c) \(\int\left(\dfrac{3}{x}-5\sqrt[3]{x}\right)dx\) (x > 0).

Answer 1

a) \(\int {\frac{1}{{{x^4}}}dx}  = \int {{x^{ - 4}}dx}  = \frac{{{x^{ - 4 + 1}}}}{{ - 4 + 1}} + C = \frac{{{x^{ - 3}}}}{{ - 3}} + C = \frac{{ - 1}}{{3{x^3}}} + C\);

b) \(\int {x\sqrt x dx = } \int {{x^{\frac{3}{2}}}dx = } \frac{{{x^{\frac{3}{2} + 1}}}}{{\frac{3}{2} + 1}} + C = \frac{2}{5}{x^2}\sqrt x  + C\);

c) \(\int {\left( {\frac{3}{x} - 5\sqrt[3]{x}} \right)dx = \int {\frac{3}{x}dx - \int {5\sqrt[3]{x}} dx = 3\int {\frac{1}{x}dx - 5\int {{x^{\frac{1}{3}}}} dx = 3\ln \left| x \right| - 5.\frac{{{x^{\frac{4}{3}}}}}{{\frac{4}{3}}} + C} } } \)

\( = 3\ln \left| x \right| - \frac{{15x\sqrt[3]{x}}}{4} + C\).