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

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

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integral of-2tan(x)

\int - 2 tan (x) d x

2 ln ∣ cos (x) ∣ + C

Solution steps

\int - 2 tan (x) d x

Take the constant out:

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

= - 2 \cdot \int tan (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2 (- ln ∣ u ∣)

Substitute back

u = cos (x)

= - 2 (- ln ∣ cos (x) ∣)

Simplify

= 2 ln ∣ cos (x) ∣

Add a constant to the solution

= 2 ln ∣ cos (x) ∣ + C

x \to + 0 lim (\frac{x ^{4}}{x ^{4}})

\int e^{5 x} cos (4 x) d x

f (x) = \frac{e ^{2 x}}{1 x + 1}

\frac{d}{d x} (\frac{x ^{2} - 4}{x ^{2} + 2})

e^{x} (5 - 4 x + 4 x^{2})