What is the integral of tan(-x) ?

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

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integral of tan(-x)

\int tan (- x) d x

ln ∣ cos (x) ∣ + C

Solution steps

\int tan (- x) d x

Apply Integration By Parts

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

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

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

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

Simplify

= ln ∣ cos (x) ∣

Add a constant to the solution

= ln ∣ cos (x) ∣ + C

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