What is the integral of sin(11x)ln(sin(11x)) ?

The integral of sin(11x)ln(sin(11x)) is 1/11 (-ln(sin(11x))cos(11x)+ln|tan((11x)/2)|+cos(11x))+C

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integral of sin(11x)ln(sin(11x))

\int sin (11 x) ln (sin (11 x)) d x

\frac{1}{11} (- ln (sin (11 x)) cos (11 x) + ln tan (\frac{11 x}{2}) + cos (11 x)) + C

Solution steps

\int sin (11 x) ln (sin (11 x)) d x

Apply u-substitution

: \int sin (u) ln (sin (u)) \frac{1}{11} d u

= \int sin (u) ln (sin (u)) \frac{1}{11} d u

Take the constant out:

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

= \frac{1}{11} \cdot \int sin (u) ln (sin (u)) d u

Apply Integration By Parts

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

= \frac{1}{11} (- ln (sin (u)) cos (u) - \int - cot (u) cos (u) d u)

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

= \frac{1}{11} (- ln (sin (u)) cos (u) - (- ln tan (\frac{u}{2}) - cos (u)))

Substitute back

u = 11 x

= \frac{1}{11} (- ln (sin (11 x)) cos (11 x) - (- ln tan (\frac{11 x}{2}) - cos (11 x)))

Simplify

= \frac{1}{11} (- ln (sin (11 x)) cos (11 x) + ln tan (\frac{11 x}{2}) + cos (11 x))

Add a constant to the solution

= \frac{1}{11} (- ln (sin (11 x)) cos (11 x) + ln tan (\frac{11 x}{2}) + cos (11 x)) + C

f (x) = x^{3} - 2 x + 1, at x = 0

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What is the integral of sin(11x)ln(sin(11x)) ?
The integral of sin(11x)ln(sin(11x)) is 1/11 (-ln(sin(11x))cos(11x)+ln|tan((11x)/2)|+cos(11x))+C