What is the integral of cot(5x) ?

The integral of cot(5x) is 1/5 ln|sin(5x)|+C

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integral of cot(5x)

\int cot (5 x) d x

\frac{1}{5} ln ∣ sin (5 x) ∣ + C

Solution steps

\int cot (5 x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 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{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = sin (5 x)

= \frac{1}{5} ln ∣ sin (5 x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ sin (5 x) ∣ + C

- \frac{12}{s ^{2} - 5}

\int 17 θ sec^{2} (θ) d θ

y = e^{c x}

\int_{0}^{4} (5 - x) x d x

x e^{y}