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

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

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

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

- 2 ln ∣ cos (x) ∣ + tan (x) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{( cos ( x ) + sin ( x ) ) ^{2}}{cos ^{2} ( x )} d x

Expand

\frac{( cos ( x ) + sin ( x ) ) ^{2}}{cos ^{2} ( x )} : 1 + \frac{2 sin ( x )}{cos ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

= \int 1 + \frac{2 sin ( x )}{cos ( x )} + \frac{sin ^{2} ( x )}{cos ^{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 \frac{2 sin ( x )}{cos ( x )} d x + \int \frac{sin ^{2} ( x )}{cos ^{2} ( x )} d x

\int 1 d x = x

\int \frac{2 sin ( x )}{cos ( x )} d x = - 2 ln ∣ cos (x) ∣

\int \frac{sin ^{2} ( x )}{cos ^{2} ( x )} d x = tan (x) - x

= x - 2 ln ∣ cos (x) ∣ + tan (x) - x

Simplify

= - 2 ln ∣ cos (x) ∣ + tan (x)

Add a constant to the solution

= - 2 ln ∣ cos (x) ∣ + tan (x) + C

y^{2} + y

x \to 3 - lim (\frac{- x}{x - 3})

ln (A x^{4} + B)

x \to - 3 lim (∣ x - 1 ∣)

\frac{\partial}{\partial x} (- 7 e^{- 7 y} sin (7 x))

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