Ολοκλήρωμα της $$$\frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}$$$ ως προς $$$x$$$

Βρείτε $$$\int \frac{1}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}\, dx$$$.

Πολλαπλασιάστε τον αριθμητή και τον παρονομαστή με $$$\sec^{2}{\left(x \right)}$$$:

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

Έστω $$$u=\tan{\left(x \right)}$$$.

Τότε $$$du=\left(\tan{\left(x \right)}\right)^{\prime }dx = \sec^{2}{\left(x \right)} dx$$$ (τα βήματα παρουσιάζονται »), και έχουμε ότι $$$\sec^{2}{\left(x \right)} dx = du$$$.

Επομένως,

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

Έστω $$$v=\frac{u \left|{b}\right|}{\left|{a}\right|}$$$.

Τότε $$$dv=\left(\frac{u \left|{b}\right|}{\left|{a}\right|}\right)^{\prime }du = \frac{\left|{b}\right|}{\left|{a}\right|} du$$$ (τα βήματα παρουσιάζονται »), και έχουμε ότι $$$du = \frac{\left|{a}\right| dv}{\left|{b}\right|}$$$.

Επομένως,

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

Εφαρμόστε τον κανόνα του σταθερού πολλαπλασίου $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ με $$$c=\frac{\left|{a}\right|}{a^{2} \left|{b}\right|}$$$ και $$$f{\left(v \right)} = \frac{1}{v^{2} + 1}$$$:

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

Το ολοκλήρωμα του $$$\frac{1}{v^{2} + 1}$$$ είναι $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$:

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

Θυμηθείτε ότι $$$v=\frac{u \left|{b}\right|}{\left|{a}\right|}$$$:

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

Θυμηθείτε ότι $$$u=\tan{\left(x \right)}$$$:

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

Επομένως,

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

Απλοποιήστε:

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

Προσθέστε τη σταθερά ολοκλήρωσης:

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

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