What is the integral of (4sin(r))/(cos(r)) ?

The integral of (4sin(r))/(cos(r)) is -4ln|cos(r)|+C

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integral of (4sin(r))/(cos(r))

\int \frac{4 sin ( r )}{cos ( r )} d r

- 4 ln ∣ cos (r) ∣ + C

Solution steps

\int \frac{4 sin ( r )}{cos ( r )} d r

Take the constant out:

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

= 4 \cdot \int \frac{sin ( r )}{cos ( r )} d r

Apply u-substitution

= 4 \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

= 4 (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 (- ln ∣ u ∣)

Substitute back

u = cos (r)

= 4 (- ln ∣ cos (r) ∣)

Simplify

= - 4 ln ∣ cos (r) ∣

Add a constant to the solution

= - 4 ln ∣ cos (r) ∣ + C

What is the integral of (4sin(r))/(cos(r)) ?
The integral of (4sin(r))/(cos(r)) is -4ln|cos(r)|+C