What is the integral of csc^3(2x+1) ?

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

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integral of csc^3(2x+1)

\int csc^{3} (2 x + 1) d x

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

Solution steps

\int csc^{3} (2 x + 1) d x

Rewrite using trig identities

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

Apply u-substitution

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

Apply u-substitution

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

Simplify

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

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

Add a constant to the solution

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

(2.3) (8.6)

\int \frac{- 4 x ^{3}}{x ^{2} + 9} d x

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

\int 12 x^{3} - 4 x d x

(1 + t) (2 t^{2} - 5)

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