What is the integral of cot(45x) ?

The integral of cot(45x) is 1/45 ln|sin(45x)|+C

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integral of cot(45x)

\int cot (45 x) d x

\frac{1}{45} ln ∣ sin (45 x) ∣ + C

Solution steps

\int cot (45 x) d x

Rewrite using trig identities

= \int \frac{cos ( 45 x )}{sin ( 45 x )} d x

Apply u-substitution

: \int \frac{1}{45 u} d u

= \int \frac{1}{45 u} d u

Take the constant out:

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

= \frac{1}{45} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{45} ln ∣ u ∣

Substitute back

u = sin (45 x)

= \frac{1}{45} ln ∣ sin (45 x) ∣

Add a constant to the solution

= \frac{1}{45} ln ∣ sin (45 x) ∣ + C

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What is the integral of cot(45x) ?
The integral of cot(45x) is 1/45 ln|sin(45x)|+C