Integral of $$$\sec^{2}{\left(\frac{x}{6} \right)}$$$

Find $$$\int \sec^{2}{\left(\frac{x}{6} \right)}\, dx$$$.

Let $$$u=\frac{x}{6}$$$.

Then $$$du=\left(\frac{x}{6}\right)^{\prime }dx = \frac{dx}{6}$$$ (steps can be seen »), and we have that $$$dx = 6 du$$$.

So,

$${\color{red}{\int{\sec^{2}{\left(\frac{x}{6} \right)} d x}}} = {\color{red}{\int{6 \sec^{2}{\left(u \right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=6$$$ and $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$:

$${\color{red}{\int{6 \sec^{2}{\left(u \right)} d u}}} = {\color{red}{\left(6 \int{\sec^{2}{\left(u \right)} d u}\right)}}$$

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

$$6 {\color{red}{\int{\sec^{2}{\left(u \right)} d u}}} = 6 {\color{red}{\tan{\left(u \right)}}}$$

Recall that $$$u=\frac{x}{6}$$$:

$$6 \tan{\left({\color{red}{u}} \right)} = 6 \tan{\left({\color{red}{\left(\frac{x}{6}\right)}} \right)}$$

Therefore,

$$\int{\sec^{2}{\left(\frac{x}{6} \right)} d x} = 6 \tan{\left(\frac{x}{6} \right)}$$

Add the constant of integration:

$$\int{\sec^{2}{\left(\frac{x}{6} \right)} d x} = 6 \tan{\left(\frac{x}{6} \right)}+C$$

$$$\int \sec^{2}{\left(\frac{x}{6} \right)}\, dx = 6 \tan{\left(\frac{x}{6} \right)} + C$$$A