What is the integral of (tan(ax)-cot(ax))^3 ?

The integral of (tan(ax)-cot(ax))^3 is -4/a (-1/2 cot^2(2ax)-ln|sin(2ax)|)+C

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integral of (tan(ax)-cot(ax))^3

\int (tan (a x) - cot (a x))^{3} d x

- \frac{4}{a} (- \frac{1}{2} cot^{2} (2 a x) - ln ∣ sin (2 a x) ∣) + C

Solution steps

\int (tan (a x) - cot (a x))^{3} d x

Simplify

(tan (a x) - cot (a x))^{3} : (tan (a x) - cot (a x))^{2} (tan (a x) - cot (a x))

= \int (tan (a x) - cot (a x))^{2} (tan (a x) - cot (a x)) d x

Apply u-substitution

: \int \frac{( tan ( u ) - cot ( u ) ) ^{3}}{a} d u

= \int \frac{( tan ( u ) - cot ( u ) ) ^{3}}{a} 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 (tan (u) - cot (u))^{3} d u

Simplify

(tan (u) - cot (u))^{3} : - 8 cot^{3} (2 u)

= \frac{1}{a} \cdot \int - 8 cot^{3} (2 u) d u

Take the constant out:

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

= \frac{1}{a} (- 8 \cdot \int cot^{3} (2 u) d u)

Apply u-substitution

: \int cot^{3} (v) \frac{1}{2} d v

= \frac{1}{a} (- 8 \cdot \int cot^{3} (v) \frac{1}{2} d v)

Take the constant out:

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

= \frac{1}{a} (- 8 \cdot \frac{1}{2} \cdot \int cot^{3} (v) d v)

Apply Integral Reduction

= \frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( v )}{2} - \int cot (v) d v))

Rewrite using trig identities

= \frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( v )}{2} - \int \frac{cos ( v )}{sin ( v )} d v))

Apply u-substitution

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

= \frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( v )}{2} - \int \frac{1}{w} d w))

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= \frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( v )}{2} - ln ∣ w ∣))

Substitute back

= \frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( 2 a x )}{2} - ln ∣ sin (2 a x) ∣))

Simplify

\frac{1}{a} (- 8 \cdot \frac{1}{2} (- \frac{cot ^{2} ( 2 a x )}{2} - ln ∣ sin (2 a x) ∣)) : - \frac{4}{a} (- \frac{1}{2} cot^{2} (2 a x) - ln ∣ sin (2 a x) ∣)

= - \frac{4}{a} (- \frac{1}{2} cot^{2} (2 a x) - ln ∣ sin (2 a x) ∣)

Add a constant to the solution

= - \frac{4}{a} (- \frac{1}{2} cot^{2} (2 a x) - ln ∣ sin (2 a x) ∣) + C

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What is the integral of (tan(ax)-cot(ax))^3 ?
The integral of (tan(ax)-cot(ax))^3 is -4/a (-1/2 cot^2(2ax)-ln|sin(2ax)|)+C