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

The integral of 2xcsc(4x)cot(4x) is 2(-1/4 xcsc(4x)+1/16 ln|tan(2x)|)+C

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integral of 2xcsc(4x)cot(4x)

\int 2 x csc (4 x) cot (4 x) d x

2 (- \frac{1}{4} x csc (4 x) + \frac{1}{16} ln ∣ tan (2 x) ∣) + C

Solution steps

\int 2 x csc (4 x) cot (4 x) d x

Take the constant out:

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

= 2 \cdot \int x csc (4 x) cot (4 x) d x

Apply Integration By Parts

= 2 (- \frac{1}{4} x csc (4 x) - \int - \frac{1}{4} csc (4 x) d x)

\int - \frac{1}{4} csc (4 x) d x = - \frac{1}{16} ln ∣ tan (2 x) ∣

= 2 (- \frac{1}{4} x csc (4 x) - (- \frac{1}{16} ln ∣ tan (2 x) ∣))

Simplify

= 2 (- \frac{1}{4} x csc (4 x) + \frac{1}{16} ln ∣ tan (2 x) ∣)

Add a constant to the solution

= 2 (- \frac{1}{4} x csc (4 x) + \frac{1}{16} ln ∣ tan (2 x) ∣) + C

What is the integral of 2xcsc(4x)cot(4x) ?
The integral of 2xcsc(4x)cot(4x) is 2(-1/4 xcsc(4x)+1/16 ln|tan(2x)|)+C