What is the integral of (cos(ax))/(1-sin(ax)) ?

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

Popular >

integral of (cos(ax))/(1-sin(ax))

\int \frac{cos ( a x )}{1 - sin ( a x )} d x

- \frac{1}{a} ln ∣ 1 - sin (a x) ∣ + C

Solution steps

\int \frac{cos ( a x )}{1 - sin ( a x )} d x

Apply u-substitution

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

= \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 = 1 - sin (a x)

= - \frac{1}{a} ln ∣ 1 - sin (a x) ∣

Add a constant to the solution

= - \frac{1}{a} ln ∣ 1 - sin (a x) ∣ + C

n = 1 \sum \infty \frac{x ^{n}}{4 n - 1}

y = e^{(4 x + 3)}

\frac{∣ x ∣}{2 - x ^{2}}

(1 + x^{2}) y^{^{'}} + 2 x y = \frac{1}{( 1 + x ^{2} ) y}

y^{^{'}} - y = 0

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