What is the integral of cos^2(x)cot(x) ?

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

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integral of cos^2(x)cot(x)

\int cos^{2} (x) cot (x) d x

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

Solution steps

\int cos^{2} (x) cot (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

= \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 (x)

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

Add a constant to the solution

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

\int \frac{x ^{2}}{x - 1} d x

x \to - 5 - lim (3 x + 15)

\frac{d}{d a} ((3 a^{2})^{3})

(x^{3} + 8 x^{3} y^{2}) d x + e^{x^{4}} y d y = 0

\int a + b x d x

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