$$$2^{5 x} 5^{- 2 x}$$$의 적분

$$$\int 2^{5 x} 5^{- 2 x}\, dx$$$을(를) 구하시오.

입력이 다음과 같이 다시 쓰입니다: $$$\int{2^{5 x} 5^{- 2 x} d x}=\int{\left(\frac{32}{25}\right)^{x} d x}$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{32}{25}$$$:

$${\color{red}{\int{\left(\frac{32}{25}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{32}{25}\right)^{x}}{\ln{\left(\frac{32}{25} \right)}}}}$$

따라서,

$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{\ln{\left(\frac{32}{25} \right)}}$$

간단히 하시오:

$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln{\left(5 \right)} + 5 \ln{\left(2 \right)}}$$

적분 상수를 추가하세요:

$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln{\left(5 \right)} + 5 \ln{\left(2 \right)}}+C$$

$$$\int 2^{5 x} 5^{- 2 x}\, dx = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln\left(5\right) + 5 \ln\left(2\right)} + C$$$A