What is the integral of cot^2(5x)csc(5x) ?

The integral of cot^2(5x)csc(5x) is 1/40 (-cot^2((5x)/2)-4ln|tan((5x)/2)|+tan^2((5x)/2))+C

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integral of cot^2(5x)csc(5x)

\int cot^{2} (5 x) csc (5 x) d x

\frac{1}{40} (- cot^{2} (\frac{5 x}{2}) - 4 ln tan (\frac{5 x}{2}) + tan^{2} (\frac{5 x}{2})) + C

Solution steps

\int cot^{2} (5 x) csc (5 x) d x

Rewrite using trig identities

= \int \frac{csc ( 5 x )}{tan ^{2} ( 5 x )} d x

Apply u-substitution

= \int \frac{csc ( u )}{5 tan ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{csc ( u )}{tan ^{2} ( u )} d u

Apply u-substitution

= \frac{1}{5} \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}{5} \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}{5} \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}{5} \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}{5} \cdot \frac{1}{4} (- \frac{1}{2 v ^{2}} - 2 ln ∣ v ∣ + \frac{v ^{2}}{2})

Substitute back

= \frac{1}{5} \cdot \frac{1}{4} (- \frac{1}{2 tan ^{2} ( \frac{5 x}{2} )} - 2 ln tan (\frac{5 x}{2}) + \frac{tan ^{2} ( \frac{5 x}{2} )}{2})

Simplify

\frac{1}{5} \cdot \frac{1}{4} (- \frac{1}{2 tan ^{2} ( \frac{5 x}{2} )} - 2 ln tan (\frac{5 x}{2}) + \frac{tan ^{2} ( \frac{5 x}{2} )}{2}) : \frac{1}{40} (- cot^{2} (\frac{5 x}{2}) - 4 ln tan (\frac{5 x}{2}) + tan^{2} (\frac{5 x}{2}))

= \frac{1}{40} (- cot^{2} (\frac{5 x}{2}) - 4 ln tan (\frac{5 x}{2}) + tan^{2} (\frac{5 x}{2}))

Add a constant to the solution

= \frac{1}{40} (- cot^{2} (\frac{5 x}{2}) - 4 ln tan (\frac{5 x}{2}) + tan^{2} (\frac{5 x}{2})) + C

What is the integral of cot^2(5x)csc(5x) ?
The integral of cot^2(5x)csc(5x) is 1/40 (-cot^2((5x)/2)-4ln|tan((5x)/2)|+tan^2((5x)/2))+C