Tìm: a) nguyên hàm (5^x+1)(5^x-1)dx; b) nguyên hàm e^0,5x dx; c) nguyên hàm 2^(x-1).5(2x+1) dx

a) \[\int {\left( {{5^x} + 1} \right)\left( {{5^x} - 1} \right)dx}  = \int {\left( {{5^{2x}} - 1} \right)dx} \]

                                   \[ = \int {{5^{2x}}dx - \int {1dx = \int {{{25}^x}dx}  - \int {1dx} } } \]

                                   \[ = \frac{{{{25}^x}}}{{\ln 25}} - x + C = \frac{{{{25}^x}}}{{2\ln 5}} - x + C.\]

b) \[\int {{e^{ - 0,5x}}dx}  = \int {{{\left( {{e^{ - 0,5}}} \right)}^x}dx = \frac{{{{\left( {{e^{ - 0,5}}} \right)}^x}}}{{\ln {e^{ - 0,5}}}}}  + C\]

                   \[ = \frac{{{e^{ - 0,5x}}}}{{ - 0,5}} + C =  - 2{e^{ - 0,5x}} + C\].

c) \[\int {{2^{x - 1}}{{.5}^{2x + 1}}dx}  = \int {\frac{{{2^x}}}{2}{{.5}^{2x}}.5dx = \int {\frac{5}{2}{{.2}^x}{{.25}^x}dx} } \]

                        \[ = \frac{5}{2}\int {{{50}^x}dx = \frac{5}{2}.\frac{{{{50}^x}}}{{\ln 50}} + C.} \]