What is the integral of cot^3(x)(sec^2(x)-1) ?

The integral of cot^3(x)(sec^2(x)-1) is ln|sin(x)|+C

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integral of cot^3(x)(sec^2(x)-1)

\int cot^{3} (x) (sec^{2} (x) - 1) d x

ln ∣ sin (x) ∣ + C

Solution steps

\int cot^{3} (x) (sec^{2} (x) - 1) d x

Rewrite using trig identities

= \int \frac{cos ( x ) ( - cos ^{2} ( x ) + 1 )}{sin ^{3} ( x )} d x

Simplify

\frac{cos ( x ) ( - cos ^{2} ( x ) + 1 )}{sin ^{3} ( x )} : cot (x)

= \int cot (x) d x

Rewrite using trig identities

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

Apply u-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 = sin (x)

= ln ∣ sin (x) ∣

Add a constant to the solution

= ln ∣ sin (x) ∣ + C

\frac{d}{d x} (\frac{100}{x})

x \to 0 + lim (x^{6} ln (x))

y^{^{'}} = \frac{x ^{2}}{y ^{2}}

\int_{1}^{e} \frac{4}{x} d x

\frac{d}{d x} (x^{3} + \frac{1}{x})

What is the integral of cot^3(x)(sec^2(x)-1) ?
The integral of cot^3(x)(sec^2(x)-1) is ln|sin(x)|+C