Integral of $$$\frac{1}{\cos^{2}{\left(x y \right)}}$$$ with respect to $$$x$$$

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

Let $$$u=x y$$$.

Then $$$du=\left(x y\right)^{\prime }dx = y dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{y}$$$.

The integral can be rewritten as

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

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

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

Rewrite the integrand in terms of the secant:

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

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

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

Recall that $$$u=x y$$$:

$$\frac{\tan{\left({\color{red}{u}} \right)}}{y} = \frac{\tan{\left({\color{red}{x y}} \right)}}{y}$$

Therefore,

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

Add the constant of integration:

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

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