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

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

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

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

- x + tan (x) + ln ∣ cos (x) ∣ + C

Solution steps

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

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

\int tan (x) d x = - ln ∣ cos (x) ∣

= - x + tan (x) - (- ln ∣ cos (x) ∣)

Simplify

= - x + tan (x) + ln ∣ cos (x) ∣

Add a constant to the solution

= - x + tan (x) + ln ∣ cos (x) ∣ + C

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