What is the integral of 5sin(x)ln(sin(x)) ?

The integral of 5sin(x)ln(sin(x)) is 5(-ln(sin(x))cos(x)+ln|tan(x/2)|+cos(x))+C

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integral of 5sin(x)ln(sin(x))

\int 5 sin (x) ln (sin (x)) d x

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

Solution steps

\int 5 sin (x) ln (sin (x)) d x

Take the constant out:

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

= 5 \cdot \int sin (x) ln (sin (x)) d x

Apply Integration By Parts

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

= 5 (- 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)

= 5 (- ln (sin (x)) cos (x) - (- ln tan (\frac{x}{2}) - cos (x)))

Simplify

= 5 (- ln (sin (x)) cos (x) + ln tan (\frac{x}{2}) + cos (x))

Add a constant to the solution

= 5 (- ln (sin (x)) cos (x) + ln tan (\frac{x}{2}) + cos (x)) + C

n = 0 \sum \infty n^{0.5}

\int (\frac{x}{x}) d x

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

\frac{d}{d x} ((tan (x)) d x)

\frac{d y}{d x} - 2 y + x^{2} = 0

What is the integral of 5sin(x)ln(sin(x)) ?
The integral of 5sin(x)ln(sin(x)) is 5(-ln(sin(x))cos(x)+ln|tan(x/2)|+cos(x))+C