What is the integral of (-y)/(cot(x)) ?

The integral of (-y)/(cot(x)) is yln|cos(x)|+C

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integral of (-y)/(cot(x))

\int \frac{- y}{cot ( x )} d x

y ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{- y}{cot ( x )} d x

Take the constant out:

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

= - y \cdot \int \frac{1}{cot ( x )} d x

Rewrite using trig identities

= - y \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int - \frac{1}{u} d u

= - y \cdot \int - \frac{1}{u} d u

Take the constant out:

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

= - y (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - y (- ln ∣ u ∣)

Substitute back

u = cos (x)

= - y (- ln ∣ cos (x) ∣)

Simplify

= y ln ∣ cos (x) ∣

Add a constant to the solution

= y ln ∣ cos (x) ∣ + C

y^{^{''}} + 2 y^{^{'}} - 24 y = 0

x y + y^{^{'}} = 76 x

\int sin (9 x) d x

\int sin (4 θ) d θ

What is the integral of (-y)/(cot(x)) ?
The integral of (-y)/(cot(x)) is yln|cos(x)|+C