What is the integral of 4xcsc^2(x) ?

The integral of 4xcsc^2(x) is 4(-xcot(x)+ln|sin(x)|)+C

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integral of 4xcsc^2(x)

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

4 (- x cot (x) + ln ∣ sin (x) ∣) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

= 4 (- x cot (x) - \int - cot (x) d x)

\int - cot (x) d x = - ln ∣ sin (x) ∣

= 4 (- x cot (x) - (- ln ∣ sin (x) ∣))

Simplify

= 4 (- x cot (x) + ln ∣ sin (x) ∣)

Add a constant to the solution

= 4 (- x cot (x) + ln ∣ sin (x) ∣) + C

What is the integral of 4xcsc^2(x) ?
The integral of 4xcsc^2(x) is 4(-xcot(x)+ln|sin(x)|)+C