What is the integral of (cos(ax))/(sin(ax)) ?

The integral of (cos(ax))/(sin(ax)) is 1/a ln|sin(ax)|+C

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integral of (cos(ax))/(sin(ax))

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

\frac{1}{a} ln ∣ sin (a x) ∣ + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{a u} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{a} ln ∣ u ∣

Substitute back

u = sin (a x)

= \frac{1}{a} ln ∣ sin (a x) ∣

Add a constant to the solution

= \frac{1}{a} ln ∣ sin (a x) ∣ + C

\int 2 x \cdot ln (x^{2}) d x

\int \frac{3}{( x - 2 ) ^{2} + 9} d x

\int \frac{1}{x ( x ^{2} + 16 )} d x

\int \frac{x + 3}{- 4 x - x ^{2}} d x

\int (- ln (x)) d x

What is the integral of (cos(ax))/(sin(ax)) ?
The integral of (cos(ax))/(sin(ax)) is 1/a ln|sin(ax)|+C