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

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{u} d u

= \int - \frac{1}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos (x)

= - ln ∣ cos (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) ∣ + C

\frac{d}{d x} (\frac{1 - e ^{x}}{1 + e ^{x}})

(ln (x^{'}))^{^{'}}

f (x) = 4 sin (2 x) + \frac{2 x ^{5} - x}{x}

\int \frac{5 x ^{2} - 1}{x ^{2}} d x

y = 5 x - x^{2}, (1, 4)