What is the integral of (8(csc(t))^2)/(cot(t)) ?

The integral of (8(csc(t))^2)/(cot(t)) is 8ln|tan(t)|+C

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integral of (8(csc(t))^2)/(cot(t))

\int \frac{8 ( csc ( t ) ) ^{2}}{cot ( t )} d t

8 ln ∣ tan (t) ∣ + C

Solution steps

\int \frac{8 ( csc ( t ) ) ^{2}}{cot ( t )} d t

Take the constant out:

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

= 8 \cdot \int \frac{( csc ( t ) ) ^{2}}{cot ( t )} d t

Rewrite using trig identities

= 8 \cdot \int \frac{sec ( t )}{sin ( t )} d t

Apply Trigonometric Substitution

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

= 8 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 ln ∣ u ∣

Substitute back

u = tan (t)

= 8 ln ∣ tan (t) ∣

Add a constant to the solution

= 8 ln ∣ tan (t) ∣ + C

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What is the integral of (8(csc(t))^2)/(cot(t)) ?
The integral of (8(csc(t))^2)/(cot(t)) is 8ln|tan(t)|+C