$$$\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}$$$ の $$$x$$$ に関する積分

$$$\int \frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}\, dx$$$ を求めよ。

分子と分母に$$$\sec^{2}{\left(x \right)}$$$を掛ける:

$${\color{red}{\int{\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{\sec^{2}{\left(x \right)}}{a^{2} + b^{2} \tan^{2}{\left(x \right)}} d x}}}$$

$$$u=\tan{\left(x \right)}$$$ とする。

すると $$$du=\left(\tan{\left(x \right)}\right)^{\prime }dx = \sec^{2}{\left(x \right)} dx$$$（手順は»で確認できます）、$$$\sec^{2}{\left(x \right)} dx = du$$$ となります。

したがって、

$${\color{red}{\int{\frac{\sec^{2}{\left(x \right)}}{a^{2} + b^{2} \tan^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{1}{a^{2} + b^{2} u^{2}} d u}}}$$

$$$v=\frac{u \left|{b}\right|}{\left|{a}\right|}$$$ とする。

すると $$$dv=\left(\frac{u \left|{b}\right|}{\left|{a}\right|}\right)^{\prime }du = \frac{\left|{b}\right|}{\left|{a}\right|} du$$$（手順は»で確認できます）、$$$du = \frac{\left|{a}\right| dv}{\left|{b}\right|}$$$ となります。

したがって、

$${\color{red}{\int{\frac{1}{a^{2} + b^{2} u^{2}} d u}}} = {\color{red}{\int{\frac{\left|{a}\right|}{a^{2} \left(v^{2} + 1\right) \left|{b}\right|} d v}}}$$

定数倍の法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ を、$$$c=\frac{\left|{a}\right|}{a^{2} \left|{b}\right|}$$$ と $$$f{\left(v \right)} = \frac{1}{v^{2} + 1}$$$ に対して適用する:

$${\color{red}{\int{\frac{\left|{a}\right|}{a^{2} \left(v^{2} + 1\right) \left|{b}\right|} d v}}} = {\color{red}{\frac{\left|{a}\right| \int{\frac{1}{v^{2} + 1} d v}}{a^{2} \left|{b}\right|}}}$$

$$$\frac{1}{v^{2} + 1}$$$ の不定積分は $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$ です:

$$\frac{\left|{a}\right| {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}}{a^{2} \left|{b}\right|} = \frac{\left|{a}\right| {\color{red}{\operatorname{atan}{\left(v \right)}}}}{a^{2} \left|{b}\right|}$$

次のことを思い出してください $$$v=\frac{u \left|{b}\right|}{\left|{a}\right|}$$$:

$$\frac{\left|{a}\right| \operatorname{atan}{\left({\color{red}{v}} \right)}}{a^{2} \left|{b}\right|} = \frac{\left|{a}\right| \operatorname{atan}{\left({\color{red}{\frac{u \left|{b}\right|}{\left|{a}\right|}}} \right)}}{a^{2} \left|{b}\right|}$$

次のことを思い出してください $$$u=\tan{\left(x \right)}$$$:

$$\frac{\left|{a}\right| \operatorname{atan}{\left(\frac{\left|{b}\right| {\color{red}{u}}}{\left|{a}\right|} \right)}}{a^{2} \left|{b}\right|} = \frac{\left|{a}\right| \operatorname{atan}{\left(\frac{\left|{b}\right| {\color{red}{\tan{\left(x \right)}}}}{\left|{a}\right|} \right)}}{a^{2} \left|{b}\right|}$$

したがって、

$$\int{\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}} d x} = \frac{\left|{a}\right| \operatorname{atan}{\left(\frac{\tan{\left(x \right)} \left|{b}\right|}{\left|{a}\right|} \right)}}{a^{2} \left|{b}\right|}$$

簡単化せよ:

$$\int{\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}} d x} = \frac{\left|{\frac{a}{b}}\right| \operatorname{atan}{\left(\tan{\left(x \right)} \left|{\frac{b}{a}}\right| \right)}}{a^{2}}$$

積分定数を加える:

$$\int{\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}} d x} = \frac{\left|{\frac{a}{b}}\right| \operatorname{atan}{\left(\tan{\left(x \right)} \left|{\frac{b}{a}}\right| \right)}}{a^{2}}+C$$

$$$\int \frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}\, dx = \frac{\left|{\frac{a}{b}}\right| \operatorname{atan}{\left(\tan{\left(x \right)} \left|{\frac{b}{a}}\right| \right)}}{a^{2}} + C$$$A