What is the integral of sin^3(3x)tan(3x) ?

The integral of sin^3(3x)tan(3x) is 1/18 (-2sin^3(3x)-6sin(3x)+3ln|sin(3x)+1|-3ln|sin(3x)-1|)+C

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integral of sin^3(3x)tan(3x)

\int sin^{3} (3 x) tan (3 x) d x

\frac{1}{18} (- 2 sin^{3} (3 x) - 6 sin (3 x) + 3 ln ∣ sin (3 x) + 1 ∣ - 3 ln ∣ sin (3 x) - 1 ∣) + C

Solution steps

\int sin^{3} (3 x) tan (3 x) d x

Apply u-substitution

: \int sin^{3} (u) tan (u) \frac{1}{3} d u

= \int sin^{3} (u) tan (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sin^{3} (u) tan (u) d u

Rewrite using trig identities

= \frac{1}{3} \cdot \int \frac{sin ^{4} ( u )}{cos ( u )} d u

Apply Trigonometric Substitution

: \int \frac{v ^{4}}{1 - v ^{2}} d v

= \frac{1}{3} \cdot \int \frac{v ^{4}}{1 - v ^{2}} d v

Long division

\frac{v ^{4}}{1 - v ^{2}} : - v^{2} - 1 + \frac{1}{- v ^{2} + 1}

= \frac{1}{3} \cdot \int - v^{2} - 1 + \frac{1}{- v ^{2} + 1} d v

Apply the Sum Rule:

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

= \frac{1}{3} (- \int v^{2} d v - \int 1 d v + \int \frac{1}{- v ^{2} + 1} d v)

\int v^{2} d v = \frac{v ^{3}}{3}

\int 1 d v = v

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{3} (- \frac{v ^{3}}{3} - v + \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

= \frac{1}{3} (- \frac{sin ^{3} ( 3 x )}{3} - sin (3 x) + \frac{ln ∣ sin ( 3 x ) + 1 ∣}{2} - \frac{ln ∣ sin ( 3 x ) - 1 ∣}{2})

Simplify

\frac{1}{3} (- \frac{sin ^{3} ( 3 x )}{3} - sin (3 x) + \frac{ln ∣ sin ( 3 x ) + 1 ∣}{2} - \frac{ln ∣ sin ( 3 x ) - 1 ∣}{2}) : \frac{1}{18} (- 2 sin^{3} (3 x) - 6 sin (3 x) + 3 ln ∣ sin (3 x) + 1 ∣ - 3 ln ∣ sin (3 x) - 1 ∣)

= \frac{1}{18} (- 2 sin^{3} (3 x) - 6 sin (3 x) + 3 ln ∣ sin (3 x) + 1 ∣ - 3 ln ∣ sin (3 x) - 1 ∣)

Add a constant to the solution

= \frac{1}{18} (- 2 sin^{3} (3 x) - 6 sin (3 x) + 3 ln ∣ sin (3 x) + 1 ∣ - 3 ln ∣ sin (3 x) - 1 ∣) + C

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What is the integral of sin^3(3x)tan(3x) ?
The integral of sin^3(3x)tan(3x) is 1/18 (-2sin^3(3x)-6sin(3x)+3ln|sin(3x)+1|-3ln|sin(3x)-1|)+C