Integral de $$$x^{2} \sec{\left(x^{3} \right)}$$$

Halla $$$\int x^{2} \sec{\left(x^{3} \right)}\, dx$$$.

Sea $$$u=x^{3}$$$.

Entonces $$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (los pasos pueden verse »), y obtenemos que $$$x^{2} dx = \frac{du}{3}$$$.

La integral se convierte en

$${\color{red}{\int{x^{2} \sec{\left(x^{3} \right)} d x}}} = {\color{red}{\int{\frac{\sec{\left(u \right)}}{3} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{3}$$$ y $$$f{\left(u \right)} = \sec{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sec{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\int{\sec{\left(u \right)} d u}}{3}\right)}}$$

Reescribe la secante como $$$\sec\left( u \right)=\frac{1}{\cos\left( u \right)}$$$:

$$\frac{{\color{red}{\int{\sec{\left(u \right)} d u}}}}{3} = \frac{{\color{red}{\int{\frac{1}{\cos{\left(u \right)}} d u}}}}{3}$$

Expresa el coseno en función del seno utilizando la fórmula $$$\cos\left( u \right)=\sin\left( u + \frac{\pi}{2}\right)$$$ y luego expresa el seno utilizando la fórmula del ángulo doble $$$\sin\left( u \right)=2\sin\left(\frac{ u }{2}\right)\cos\left(\frac{ u }{2}\right)$$$:

$$\frac{{\color{red}{\int{\frac{1}{\cos{\left(u \right)}} d u}}}}{3} = \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{u}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{u}{2} + \frac{\pi}{4} \right)}} d u}}}}{3}$$

Multiplica el numerador y el denominador por $$$\sec^2\left(\frac{ u }{2} + \frac{\pi}{4} \right)$$$:

$$\frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{u}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{u}{2} + \frac{\pi}{4} \right)}} d u}}}}{3} = \frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}} d u}}}}{3}$$

Sea $$$v=\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}$$$.

Entonces $$$dv=\left(\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}\right)^{\prime }du = \frac{\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)}}{2} du$$$ (los pasos pueden verse »), y obtenemos que $$$\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)} du = 2 dv$$$.

Por lo tanto,

$$\frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}} d u}}}}{3} = \frac{{\color{red}{\int{\frac{1}{v} d v}}}}{3}$$

La integral de $$$\frac{1}{v}$$$ es $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{v} d v}}}}{3} = \frac{{\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{3}$$

Recordemos que $$$v=\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}$$$:

$$\frac{\ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{3} = \frac{\ln{\left(\left|{{\color{red}{\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}}}}\right| \right)}}{3}$$

Recordemos que $$$u=x^{3}$$$:

$$\frac{\ln{\left(\left|{\tan{\left(\frac{\pi}{4} + \frac{{\color{red}{u}}}{2} \right)}}\right| \right)}}{3} = \frac{\ln{\left(\left|{\tan{\left(\frac{\pi}{4} + \frac{{\color{red}{x^{3}}}}{2} \right)}}\right| \right)}}{3}$$

Por lo tanto,

$$\int{x^{2} \sec{\left(x^{3} \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x^{3}}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{3}$$

Añade la constante de integración:

$$\int{x^{2} \sec{\left(x^{3} \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x^{3}}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{3}+C$$

$$$\int x^{2} \sec{\left(x^{3} \right)}\, dx = \frac{\ln\left(\left|{\tan{\left(\frac{x^{3}}{2} + \frac{\pi}{4} \right)}}\right|\right)}{3} + C$$$A