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

Take the constant out:

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

= - \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

x \to 4 - lim (\frac{2 - x}{x - 4})

arcsin (sin (x))

\frac{d}{d x} (\frac{1}{x ln ( 10 )})

f (x) = \frac{1 + e ^{x}}{1 - e ^{x}}

\frac{d y}{d x} = e^{(3 x + 2 y)}

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