What is the integral of arcot(x) ?

The integral of arcot(x) is arln|sin(x)|+C

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integral of arcot(x)

\int a r cot (x) d x

a r ln ∣ sin (x) ∣ + C

Solution steps

\int a r cot (x) d x

Take the constant out:

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

= a r \cdot \int cot (x) d x

Rewrite using trig identities

= a r \cdot \int \frac{cos ( x )}{sin ( x )} d x

Apply u-substitution

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

= a r \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a r ln ∣ u ∣

Substitute back

u = sin (x)

= a r ln ∣ sin (x) ∣

Add a constant to the solution

= a r ln ∣ sin (x) ∣ + C

\frac{d}{d x} ((\frac{e}{x})^{x})

x \to 0 lim (\frac{( 5 ^{x} - 1 )}{x})

x \to 0 lim (3^{x} + 7)

\frac{d}{d x} (x lo g_{e} (x) - x)

\int x (x + 5)^{8} d x