What is the integral of sin(2x)ln(sin(2x)) ?

The integral of sin(2x)ln(sin(2x)) is -1/2 ln(sin(2x))cos(2x)+1/2 ln|tan(x)|+1/2 cos(2x)+C

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integral of sin(2x)ln(sin(2x))

\int sin (2 x) ln (sin (2 x)) d x

- \frac{1}{2} ln (sin (2 x)) cos (2 x) + \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x) + C

Solution steps

\int sin (2 x) ln (sin (2 x)) d x

Apply Integration By Parts

: - \frac{1}{2} ln (sin (2 x)) cos (2 x) - \int - cot (2 x) cos (2 x) d x

= - \frac{1}{2} ln (sin (2 x)) cos (2 x) - \int - cot (2 x) cos (2 x) d x

\int - cot (2 x) cos (2 x) d x = - \frac{1}{2} ln ∣ tan (x) ∣ - \frac{1}{2} cos (2 x)

= - \frac{1}{2} ln (sin (2 x)) cos (2 x) - (- \frac{1}{2} ln ∣ tan (x) ∣ - \frac{1}{2} cos (2 x))

Simplify

= - \frac{1}{2} ln (sin (2 x)) cos (2 x) + \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x)

Add a constant to the solution

= - \frac{1}{2} ln (sin (2 x)) cos (2 x) + \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} cos (2 x) + C

\frac{10}{s ( s ^{2} + 2 s + 10 )}

\frac{\partial}{\partial y} (x y + y^{2} + y)

f (x) = \frac{2}{x}, at (\frac{1}{9})

\int x arctan (4 x) d x

x^{2} - 4 x + 1

What is the integral of sin(2x)ln(sin(2x)) ?
The integral of sin(2x)ln(sin(2x)) is -1/2 ln(sin(2x))cos(2x)+1/2 ln|tan(x)|+1/2 cos(2x)+C