Evaluate the following integral. \int (5+x+\tan^{2}x)dx

Step 1: Expand.
${\displaystyle \int 5+x+{\tan }^{2}xdx}$
Step 2: Use Sum Rule: ${\displaystyle \int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx}$.
${\displaystyle \int 5+xdx+\int {\tan }^{2}xdx}$
Step 3: Use Power Rule: ${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$.
${\displaystyle 5x+\frac{x^2}{2}+\int {\tan }^{2}xdx}$
Step 4: Use Pythagorean Identities: ${\displaystyle {\tan }^{2}x={\sec }^{2}x-1}$.
${\displaystyle 5x+\frac{x^2}{2}+\int {\sec }^{2}x-1dx}$
Step 5: Use Sum Rule: ${\displaystyle \int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx}$.
${\displaystyle 5x+\frac{x^2}{2}+\int {\sec }^{2}xdx+\int -1dx}$
Step 6: The derivative of \tan x is ${\displaystyle {\sec }^{2}x}$.
${\displaystyle 5x+\frac{x^2}{2}+\tan x+\int -1dx}$
Step 7: Use this rule: ${\displaystyle \int adx=ax+C}$.
${\displaystyle 4x+\frac{x^2}{2}+\tan x}$
Step 8: Add constant.
${\displaystyle 4x+\frac{x^2}{2}+\tan x+C}$