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

The integral of csc(5x)cot(5x) is -1/5 csc(5x)+C

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

\int csc (5 x) cot (5 x) d x

- \frac{1}{5} csc (5 x) + C

Solution steps

\int csc (5 x) cot (5 x) d x

Rewrite using trig identities

= \int \frac{cot ( 5 x )}{sin ( 5 x )} d x

Apply u-substitution

: \int \frac{cot ( u )}{5 sin ( u )} d u

= \int \frac{cot ( u )}{5 sin ( 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{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

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

= \frac{1}{5} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{5} (- \frac{1}{v})

Substitute back

= \frac{1}{5} (- \frac{1}{sin ( 5 x )})

Simplify

\frac{1}{5} (- \frac{1}{sin ( 5 x )}) : - \frac{1}{5} csc (5 x)

= - \frac{1}{5} csc (5 x)

Add a constant to the solution

= - \frac{1}{5} csc (5 x) + C

\int_{2}^{5} \frac{5}{x - 2} d x

\frac{d}{d y} (2 x (y))

\frac{d}{d x} (\frac{( - x ^{2} - 5 )}{( x ^{2} - 5 ) ^{2}})

- 2 e^{3 t}

25 y^{^{''}} - 25 y^{^{'}} - 6 y = 0

What is the integral of csc(5x)cot(5x) ?
The integral of csc(5x)cot(5x) is -1/5 csc(5x)+C