$$$2^{- \frac{t}{5}}$$$ 的積分
您的輸入
求$$$\int 2^{- \frac{t}{5}}\, dt$$$。
解答
已將輸入重寫為：$$$\int{2^{- \frac{t}{5}} d t}=\int{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t} d t}$$$。
Apply the exponential rule $$$\int{a^{t} d t} = \frac{a^{t}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{2^{\frac{4}{5}}}{2}$$$:
$${\color{red}{\int{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t} d t}}} = {\color{red}{\frac{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t}}{\ln{\left(\frac{2^{\frac{4}{5}}}{2} \right)}}}}$$
因此，
$$\int{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t} d t} = \frac{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t}}{\ln{\left(\frac{2^{\frac{4}{5}}}{2} \right)}}$$
化簡：
$$\int{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t} d t} = - \frac{5 \cdot 2^{- \frac{t}{5}}}{\ln{\left(2 \right)}}$$
加上積分常數：
$$\int{\left(\frac{2^{\frac{4}{5}}}{2}\right)^{t} d t} = - \frac{5 \cdot 2^{- \frac{t}{5}}}{\ln{\left(2 \right)}}+C$$
答案
$$$\int 2^{- \frac{t}{5}}\, dt = - \frac{5 \cdot 2^{- \frac{t}{5}}}{\ln\left(2\right)} + C$$$A