What is the integral of (1+tan(x))^3 ?

The integral of (1+tan(x))^3 is -2x-3ln|cos(x)|+3tan(x)-ln|sec(x)|+1/2 sec^2(x)+C

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integral of (1+tan(x))^3

\int (1 + tan (x))^{3} d x

- 2 x - 3 ln ∣ cos (x) ∣ + 3 tan (x) - ln ∣ sec (x) ∣ + \frac{1}{2} sec^{2} (x) + C

Solution steps

\int (1 + tan (x))^{3} d x

Expand

(1 + tan (x))^{3} : 1 + 3 tan (x) + 3 tan^{2} (x) + tan^{3} (x)

= \int 1 + 3 tan (x) + 3 tan^{2} (x) + tan^{3} (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 3 tan (x) d x + \int 3 tan^{2} (x) d x + \int tan^{3} (x) d x

\int 1 d x = x

\int 3 tan (x) d x = - 3 ln ∣ cos (x) ∣

\int 3 tan^{2} (x) d x = 3 (- x + tan (x))

\int tan^{3} (x) d x = - ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}

= x - 3 ln ∣ cos (x) ∣ + 3 (- x + tan (x)) - ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}

Simplify

x - 3 ln ∣ cos (x) ∣ + 3 (- x + tan (x)) - ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2} : - 2 x - 3 ln ∣ cos (x) ∣ + 3 tan (x) - ln ∣ sec (x) ∣ + \frac{1}{2} sec^{2} (x)

= - 2 x - 3 ln ∣ cos (x) ∣ + 3 tan (x) - ln ∣ sec (x) ∣ + \frac{1}{2} sec^{2} (x)

Add a constant to the solution

= - 2 x - 3 ln ∣ cos (x) ∣ + 3 tan (x) - ln ∣ sec (x) ∣ + \frac{1}{2} sec^{2} (x) + C

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What is the integral of (1+tan(x))^3 ?
The integral of (1+tan(x))^3 is -2x-3ln|cos(x)|+3tan(x)-ln|sec(x)|+1/2 sec^2(x)+C