What is the integral of (-sin(x))/(cos(x)) ?

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

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integral of (-sin(x))/(cos(x))

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

ln ∣ cos (x) ∣ + C

Solution steps

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

Take the constant out:

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

= - \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) ∣)

Simplify

= ln ∣ cos (x) ∣

Add a constant to the solution

= ln ∣ cos (x) ∣ + C

\int \frac{2 x}{x ^{2} - 4} d x

\int 2 e^{- 2 x} sin (2 x) d x

\frac{d y}{d x} - 2 y tan (x) = y^{2} tan^{2} (x)

\int \frac{x ^{2}}{x - 6} d x

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

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