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

The integral of (2+tan(x))^2 is 3x-4ln|cos(x)|+tan(x)+C

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

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

3 x - 4 ln ∣ cos (x) ∣ + tan (x) + C

Solution steps

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

Expand

(2 + tan (x))^{2} : 4 + 4 tan (x) + tan^{2} (x)

= \int 4 + 4 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 4 d x + \int 4 tan (x) d x + \int tan^{2} (x) d x

\int 4 d x = 4 x

\int 4 tan (x) d x = - 4 ln ∣ cos (x) ∣

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

= 4 x - 4 ln ∣ cos (x) ∣ - x + tan (x)

Simplify

= 3 x - 4 ln ∣ cos (x) ∣ + tan (x)

Add a constant to the solution

= 3 x - 4 ln ∣ cos (x) ∣ + tan (x) + C

f (x) = 4 sec^{2} (x)

f (x) = 5 x 1 - x^{3}

\frac{d}{d x} (\frac{7 x + 8}{4 x + 8})

\int - t e^{- 2 t} d t

t^{2} \frac{d y}{d t} + y = 0, y (5) = - 1

What is the integral of (2+tan(x))^2 ?
The integral of (2+tan(x))^2 is 3x-4ln|cos(x)|+tan(x)+C