What is the integral of (4(csc(x))^2)/(cot(x)) ?

The integral of (4(csc(x))^2)/(cot(x)) is 4ln|tan(x)|+C

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integral of (4(csc(x))^2)/(cot(x))

\int \frac{4 ( csc ( x ) ) ^{2}}{cot ( x )} d x

4 ln ∣ tan (x) ∣ + C

Solution steps

\int \frac{4 ( csc ( x ) ) ^{2}}{cot ( x )} d x

Take the constant out:

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

= 4 \cdot \int \frac{( csc ( x ) ) ^{2}}{cot ( x )} d x

Rewrite using trig identities

= 4 \cdot \int \frac{sec ( x )}{sin ( x )} d x

Apply Trigonometric Substitution

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

= 4 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 4 ln ∣ u ∣

Substitute back

u = tan (x)

= 4 ln ∣ tan (x) ∣

Add a constant to the solution

= 4 ln ∣ tan (x) ∣ + C

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What is the integral of (4(csc(x))^2)/(cot(x)) ?
The integral of (4(csc(x))^2)/(cot(x)) is 4ln|tan(x)|+C