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

The integral of (cos^3(t))/(sin(t)) is ln|sin(t)|-(sin^2(t))/2+C

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integral of (cos^3(t))/(sin(t))

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

ln ∣ sin (t) ∣ - \frac{sin ^{2} ( t )}{2} + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

= \int \frac{1 - u ^{2}}{u} d u

Expand

\frac{1 - u ^{2}}{u} : \frac{1}{u} - u

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= ln ∣ u ∣ - \frac{u ^{2}}{2}

Substitute back

u = sin (t)

= ln ∣ sin (t) ∣ - \frac{sin ^{2} ( t )}{2}

Add a constant to the solution

= ln ∣ sin (t) ∣ - \frac{sin ^{2} ( t )}{2} + C

What is the integral of (cos^3(t))/(sin(t)) ?
The integral of (cos^3(t))/(sin(t)) is ln|sin(t)|-(sin^2(t))/2+C