What is the integral of (1+tan(x))/(1-tan(x)) ?

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

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integral of (1+tan(x))/(1-tan(x))

\int \frac{1 + tan ( x )}{1 - tan ( x )} d x

- ln ∣ cos (x) - sin (x) ∣ + C

Solution steps

\int \frac{1 + tan ( x )}{1 - tan ( x )} d x

Rewrite using trig identities

= \int \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( 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) - sin (x)

= - ln ∣ cos (x) - sin (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) - sin (x) ∣ + C

x \to \infty lim ((1 - \frac{1}{4 x})^{8 x})

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\int (5 x - 1) (3 x^{2} + 2) d x

\int e^{x^{3}} (12 x^{2}) d x

\frac{\partial}{\partial x} (x e^{- \frac{y}{x}})

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