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

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

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integral of (tan(2x))

\int (tan (2 x)) d x

- \frac{1}{2} ln ∣ cos (2 x) ∣ + C

Solution steps

\int tan (2 x) d x

Rewrite using trig identities

= \int \frac{sin ( 2 x )}{cos ( 2 x )} d x

Apply u-substitution

: \int - \frac{1}{2 u} d u

= \int - \frac{1}{2 u} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = cos (2 x)

= - \frac{1}{2} ln ∣ cos (2 x) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ cos (2 x) ∣ + C

\frac{d}{d x} (\frac{1}{8} (- x^{2} + 8 ln (x)))

y^{^{''}} + 6 y^{^{'}} + 25 y = 0, y (0) = 2, y^{^{'}} (0) = - 5

\int_{0}^{7} k (7 - x)

t \to + 0 lim (18 (1.15)^{t})

x \to 7 + lim (\frac{9 x}{x - 7})

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