What is the integral of 8x*csc^2(4x) ?

The integral of 8x*csc^2(4x) is 8(-1/4 xcot(4x)+1/16 ln|sin(4x)|)+C

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integral of 8x*csc^2(4x)

\int 8 x \cdot csc^{2} (4 x) d x

8 (- \frac{1}{4} x cot (4 x) + \frac{1}{16} ln ∣ sin (4 x) ∣) + C

Solution steps

\int 8 x csc^{2} (4 x) d x

Take the constant out:

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

= 8 \cdot \int x csc^{2} (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} x cot (4 x) - \int - \frac{1}{4} cot (4 x) d x

= 8 (- \frac{1}{4} x cot (4 x) - \int - \frac{1}{4} cot (4 x) d x)

\int - \frac{1}{4} cot (4 x) d x = - \frac{1}{16} ln ∣ sin (4 x) ∣

= 8 (- \frac{1}{4} x cot (4 x) - (- \frac{1}{16} ln ∣ sin (4 x) ∣))

Simplify

= 8 (- \frac{1}{4} x cot (4 x) + \frac{1}{16} ln ∣ sin (4 x) ∣)

Add a constant to the solution

= 8 (- \frac{1}{4} x cot (4 x) + \frac{1}{16} ln ∣ sin (4 x) ∣) + C

\int 2^{5 x + 2} d x

2 x y^{^{'}} + y = 6 x, y (4) = 15

\int \frac{x ^{'}}{x ^{2}} d x

\int tan^{2} (2 x) d x

What is the integral of 8x*csc^2(4x) ?
The integral of 8x*csc^2(4x) is 8(-1/4 xcot(4x)+1/16 ln|sin(4x)|)+C