What is the integral of-3cot(t) ?

The integral of-3cot(t) is -3ln|sin(t)|+C

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integral of-3cot(t)

\int - 3 cot (t) d t

- 3 ln ∣ sin (t) ∣ + C

Solution steps

\int - 3 cot (t) d t

Take the constant out:

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

= - 3 \cdot \int cot (t) d t

Rewrite using trig identities

= - 3 \cdot \int \frac{cos ( t )}{sin ( t )} d t

Apply u-substitution

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

= - 3 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 3 ln ∣ u ∣

Substitute back

u = sin (t)

= - 3 ln ∣ sin (t) ∣

Add a constant to the solution

= - 3 ln ∣ sin (t) ∣ + C

\int (\frac{x + 5}{x ^{2} + x - 6}) d x

\int \frac{4 x}{x ^{2} - 1} d x

f (x) = arctan (\frac{x}{2})

\int \frac{x}{x ^{4} - 3 x ^{2} + 2} d x

(sin (\frac{1}{x}))^{^{'}}