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

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

Popular >

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

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

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

Solution steps

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

Apply Integration By Parts

: - ln (sin (x)) cos (x) - \int - cot (x) cos (x) d x

= - ln (sin (x)) cos (x) - \int - cot (x) cos (x) d x

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

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

Simplify

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

Add a constant to the solution

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

x^{6}, at x = 3

x \to 0 lim (\frac{x}{x - 1})

\frac{d y}{d x} + y = e^{3 x}

\frac{d y}{d x} - y = 3 e^{8 x}

\frac{cos ( x )}{x ^{4}}

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