What is the integral of csc^3(2x) ?

The integral of csc^3(2x) is 1/16 (-cot^2(x)+4ln|tan(x)|+tan^2(x))+C

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integral of csc^3(2x)

\int csc^{3} (2 x) d x

\frac{1}{16} (- cot^{2} (x) + 4 ln ∣ tan (x) ∣ + tan^{2} (x)) + C

Solution steps

\int csc^{3} (2 x) d x

Apply u-substitution

: \int csc^{3} (u) \frac{1}{2} d u

= \int csc^{3} (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int csc^{3} (u) d u

Apply u-substitution

: \int \frac{( 1 + v ^{2} ) ^{2}}{4 v ^{3}} d v

= \frac{1}{2} \cdot \int \frac{( 1 + v ^{2} ) ^{2}}{4 v ^{3}} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int \frac{( 1 + v ^{2} ) ^{2}}{v ^{3}} d v

Expand

\frac{( 1 + v ^{2} ) ^{2}}{v ^{3}} : \frac{1}{v ^{3}} + \frac{2}{v} + v

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int \frac{1}{v ^{3}} + \frac{2}{v} + v 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}{2} \cdot \frac{1}{4} (\int \frac{1}{v ^{3}} d v + \int \frac{2}{v} d v + \int v d v)

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

\int \frac{2}{v} d v = 2 ln ∣ v ∣

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

= \frac{1}{2} \cdot \frac{1}{4} (- \frac{1}{2 v ^{2}} + 2 ln ∣ v ∣ + \frac{v ^{2}}{2})

Substitute back

= \frac{1}{2} \cdot \frac{1}{4} (- \frac{1}{2 tan ^{2} ( \frac{2 x}{2} )} + 2 ln tan (\frac{2 x}{2}) + \frac{tan ^{2} ( \frac{2 x}{2} )}{2})

Simplify

\frac{1}{2} \cdot \frac{1}{4} (- \frac{1}{2 tan ^{2} ( \frac{2 x}{2} )} + 2 ln tan (\frac{2 x}{2}) + \frac{tan ^{2} ( \frac{2 x}{2} )}{2}) : \frac{1}{16} (- cot^{2} (x) + 4 ln ∣ tan (x) ∣ + tan^{2} (x))

= \frac{1}{16} (- cot^{2} (x) + 4 ln ∣ tan (x) ∣ + tan^{2} (x))

Add a constant to the solution

= \frac{1}{16} (- cot^{2} (x) + 4 ln ∣ tan (x) ∣ + tan^{2} (x)) + C

x \to 0 lim (sin (5 x) csc (9 x))

\frac{( s + 1 ) ^{3}}{s ^{4}}

y = \frac{e ^{\frac{x}{2}}}{x + 1}

\frac{1}{x ^{6}}

\int_{- \infty}^{0} x e^{3 x} d x

What is the integral of csc^3(2x) ?
The integral of csc^3(2x) is 1/16 (-cot^2(x)+4ln|tan(x)|+tan^2(x))+C