Question

Tìm: 

a) \(\int\left(3\sqrt{x}+\dfrac{1}{\sqrt[3]{x}}\right)dx;\)                       b) \(\int\left(7x^2-3\right)dx\) (x > 0);

c) \(\int\dfrac{\left(2x+1\right)^2}{x^2}dx;\)                                d) \(\int\left(2^x+\dfrac{3}{x^2}\right)dx.\)

Answer 1

a) \(\int {\left( {3\sqrt x  + \frac{1}{{\sqrt[3]{x}}}} \right)} dx = 3\int {{x^{\frac{1}{2}}}} dx + \int {{x^{\frac{{ - 1}}{3}}}} dx = 2x\sqrt x  + \frac{3}{2}\sqrt[3]{{{x^2}}} + C\)

b) \(\int {\sqrt x \left( {7{x^2} - 3} \right)} dx = \int {\left( {7{x^{\frac{5}{2}}} - 3{x^{\frac{1}{2}}}} \right)dx = } 7\int {{x^{\frac{5}{2}}}} dx - 3\int {{x^{\frac{1}{2}}}} dx = 2{x^3}\sqrt x  - 2x\sqrt x  + C\)

c) \(\int {\frac{{{{\left( {2x + 1} \right)}^2}}}{{{x^2}}}} dx = \int {\frac{{4{x^2} + 4x + 1}}{{{x^2}}}} dx = \int 4 dx + 4\int {\frac{1}{x}} dx + \int {{x^{ - 2}}} dx = 4x + 4\ln \left| x \right| - \frac{1}{x} + C\)

d) \(\int {\left( {{2^x} + \frac{3}{{{x^2}}}} \right)} dx = \int {{2^x}} dx + 3\int {{x^{ - 2}}} dx = \frac{{{2^x}}}{{\ln 2}} - \frac{3}{x} + C\)