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$$$\left(\frac{11}{5}\right)^{x}$$$の積分
入力内容
$$$\int \left(\frac{11}{5}\right)^{x}\, dx$$$ を求めよ。
解答
Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{11}{5}$$$:
$${\color{red}{\int{\left(\frac{11}{5}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{11}{5}\right)^{x}}{\ln{\left(\frac{11}{5} \right)}}}}$$
したがって、
$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{\ln{\left(\frac{11}{5} \right)}}$$
簡単化せよ:
$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln{\left(5 \right)} + \ln{\left(11 \right)}}$$
積分定数を加える:
$$\int{\left(\frac{11}{5}\right)^{x} d x} = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln{\left(5 \right)} + \ln{\left(11 \right)}}+C$$
解答
$$$\int \left(\frac{11}{5}\right)^{x}\, dx = \frac{\left(\frac{11}{5}\right)^{x}}{- \ln\left(5\right) + \ln\left(11\right)} + C$$$A