What is the integral of (1+sec(x))^2 ?

The integral of (1+sec(x))^2 is x+2ln|tan(x)+sec(x)|+tan(x)+C

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integral of (1+sec(x))^2

\int (1 + sec (x))^{2} d x

x + 2 ln ∣ tan (x) + sec (x) ∣ + tan (x) + C

Solution steps

\int (1 + sec (x))^{2} d x

Expand

(1 + sec (x))^{2} : 1 + 2 sec (x) + sec^{2} (x)

= \int 1 + 2 sec (x) + sec^{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 1 d x + \int 2 sec (x) d x + \int sec^{2} (x) d x

\int 1 d x = x

\int 2 sec (x) d x = 2 ln ∣ tan (x) + sec (x) ∣

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

= x + 2 ln ∣ tan (x) + sec (x) ∣ + tan (x)

Add a constant to the solution

= x + 2 ln ∣ tan (x) + sec (x) ∣ + tan (x) + C

\frac{x}{cos ( x )}

x^{3} - 6 x^{2} + 9 x + 2

x^{2 c o s (x)}

x = 4 sin (t)

y = - 30 \frac{ln ( \frac{1}{40} )}{ln ( 2 )}

What is the integral of (1+sec(x))^2 ?
The integral of (1+sec(x))^2 is x+2ln|tan(x)+sec(x)|+tan(x)+C