Integral of $$$\frac{1}{4 \cos^{2}{\left(x \right)}}$$$

Find $$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$$$:

$${\color{red}{\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}{4}\right)}}$$

Rewrite the integrand in terms of the secant:

$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}}}{4} = \frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4}$$

The integral of $$$\sec^{2}{\left(x \right)}$$$ is $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

$$\frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4} = \frac{{\color{red}{\tan{\left(x \right)}}}}{4}$$

Therefore,

$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}$$

Add the constant of integration:

$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}+C$$

$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx = \frac{\tan{\left(x \right)}}{4} + C$$$A