Giải bài 24 trang 15

Tìm: a) (int {{e^{5x}}} dx); b) (int {frac{1}{{{{2024}^x}}}} dx); c) (int {left( {{2^x} + {x^2}} right)} dx); d) (int {left( {{2^x}{{.3}^{2{rm{x}} + 1}}} right)} dx); e) (int {frac{{{3^x} + {4^x} + 1}}{{{5^x}}}} dx).

Đề bài

Tìm:

a) $\int e^{5 x} d x$ ;

b) $\int \frac{1}{2024^{x}} d x$ ;

c) $\int (2^{x} + x^{2}) d x$ ;

d) $\int (2^{x} {.3}^{2 x + 1}) d x$ ;

e) $\int \frac{3^{x} + 4^{x} + 1}{5^{x}} d x$ .

Phương pháp giải - Xem chi tiết

Sử dụng các công thức:

• $\int e^{x} d x = e^{x} + C$ .

• $\int a^{x} d x = \frac{a^{x}}{\ln a} + C$ .

• $\int x^{α} d x = \frac{x^{α + 1}}{α + 1} + C$ .

Lời giải chi tiết

a) $\int e^{5 x} d x = \int {(e^{5})}^{x} d x = \frac{{(e^{5})}^{x}}{\ln e^{5}} + C = \frac{e^{5 x}}{5} + C$ .

b) $\int \frac{1}{2024^{x}} d x = \int {(\frac{1}{2024})}^{x} d x$

$= \frac{{(\frac{1}{2024})}^{x}}{\ln \frac{1}{2024}} + C = - \frac{1}{2024^{x} \ln 2024} + C$ .

c) $\int (2^{x} + x^{2}) d x = \frac{2^{x}}{\ln 2} + \frac{x^{3}}{3} + C$ .

d) $\int (2^{x} {.3}^{2 x + 1}) d x = \int (2^{x} . {(3^{2})}^{x} .3) d x$

$= \int (2^{x} {.9}^{x} .3) d x = 3 \int {(2.9)}^{x} d x$

$= 3 \int 18^{x} d x = 3. \frac{18^{x}}{\ln 18} + C$ .

e) $\int \frac{3^{x} + 4^{x} + 1}{5^{x}} d x = \int (\frac{3^{x}}{5^{x}} + \frac{4^{x}}{5^{x}} + \frac{1}{5^{x}}) d x$

$= \int ({(\frac{3}{5})}^{x} + {(\frac{4}{5})}^{x} + {(\frac{1}{5})}^{x}) d x$

$= \frac{{(\frac{3}{5})}^{x}}{\ln \frac{3}{5}} + \frac{{(\frac{4}{5})}^{x}}{\ln \frac{4}{5}} + \frac{{(\frac{1}{5})}^{x}}{\ln \frac{1}{5}} + C$

$= \frac{3^{x}}{5^{x} (\ln 3 - \ln 5)} + \frac{4^{x}}{5^{x} (\ln 4 - \ln 5)} - \frac{1}{5^{x} \ln 5} + C$ .