What is the integral of (sec(x)+tan(x))^2 ?

The integral of (sec(x)+tan(x))^2 is -x+2tan(x)+2sec(x)+C

integral of (sec(x)+tan(x))^2

\int (sec (x) + tan (x))^{2} d x

- x + 2 tan (x) + 2 sec (x) + C

Solution steps

\int (sec (x) + tan (x))^{2} d x

Expand

(sec (x) + tan (x))^{2} : sec^{2} (x) + 2 sec (x) tan (x) + tan^{2} (x)

= \int sec^{2} (x) + 2 sec (x) tan (x) + tan^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec^{2} (x) d x + \int 2 sec (x) tan (x) d x + \int tan^{2} (x) d x

\int sec^{2} (x) d x = tan (x)

\int 2 sec (x) tan (x) d x = 2 sec (x)

\int tan^{2} (x) d x = - x + tan (x)

= tan (x) + 2 sec (x) - x + tan (x)

Simplify

= - x + 2 tan (x) + 2 sec (x)

Add a constant to the solution

= - x + 2 tan (x) + 2 sec (x) + C

\frac{d ^{2} y}{d x ^{2}} = 2 x - 3

x \to 0 lim (x^{4} sin (\frac{3}{x}))

\frac{d}{d x} ((2 x^{3} + 5)^{4})

\int (8 t^{4} + 5 t^{3} - 6 t^{2} + 7 t - 9) d t

\int \frac{3}{x ^{5}} d x

What is the integral of (sec(x)+tan(x))^2 ?
The integral of (sec(x)+tan(x))^2 is -x+2tan(x)+2sec(x)+C