What is the integral of-cot(2x) ?

The integral of-cot(2x) is -1/2 ln|sin(2x)|+C

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integral of-cot(2x)

\int - cot (2 x) d x

- \frac{1}{2} ln ∣ sin (2 x) ∣ + C

Solution steps

\int - cot (2 x) d x

Take the constant out:

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

= - \int cot (2 x) d x

Rewrite using trig identities

= - \int \frac{cos ( 2 x )}{sin ( 2 x )} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= - \int \frac{1}{2 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}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = sin (2 x)

= - \frac{1}{2} ln ∣ sin (2 x) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ sin (2 x) ∣ + C

\int x (a - x)^{2} d x

\frac{d}{d x} (\frac{2}{x ^{2} + 1})

\int_{0}^{2} 2 π (5 - x) (8 - x^{3}) d x

\int (\frac{4}{x ^{2}}) d x

\frac{x d y}{d x} + 5 y = 7 x^{2}, f (2) = 5