What is the integral of (sin(x))/(cos^3(x)-cos(x)) ?

The integral of (sin(x))/(cos^3(x)-cos(x)) is -ln|tan(x)|+C

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integral of (sin(x))/(cos^3(x)-cos(x))

\int \frac{sin ( x )}{cos ^{3} ( x ) - cos ( x )} d x

- ln ∣ tan (x) ∣ + C

Solution steps

\int \frac{sin ( x )}{cos ^{3} ( x ) - cos ( x )} d x

Factor

cos^{3} (x) - cos (x) : cos (x) (cos^{2} (x) - 1)

= \int \frac{sin ( x )}{cos ( x ) ( cos ^{2} ( x ) - 1 )} d x

Rewrite using trig identities

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

Take the constant out:

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

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

Rewrite using trig identities

= - \int \frac{sec ( x )}{sin ( x )} d x

Apply Trigonometric Substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = tan (x)

= - ln ∣ tan (x) ∣

Add a constant to the solution

= - ln ∣ tan (x) ∣ + C

\int \frac{3 + x}{x ^{2023}} d x

x + 1

\int (\frac{1}{2} x^{4} + \frac{1}{4} x^{3} - x) d x

\int 2 x - 0.25 x^{2} d x

\int \frac{2 y}{y ^{2} - 5} d y

What is the integral of (sin(x))/(cos^3(x)-cos(x)) ?
The integral of (sin(x))/(cos^3(x)-cos(x)) is -ln|tan(x)|+C