What is the integral of sin(x)ln(1+sin(x)) ?

The integral of sin(x)ln(1+sin(x)) is -ln(1+sin(x))cos(x)+x+cos(x)+C

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integral of sin(x)ln(1+sin(x))

\int sin (x) ln (1 + sin (x)) d x

- ln (1 + sin (x)) cos (x) + x + cos (x) + C

Solution steps

\int sin (x) ln (1 + sin (x)) d x

Apply Integration By Parts

: - ln (1 + sin (x)) cos (x) - \int - 1 + sin (x) d x

= - ln (1 + sin (x)) cos (x) - \int - 1 + sin (x) d x

\int - 1 + sin (x) d x = - x - cos (x)

= - ln (1 + sin (x)) cos (x) - (- x - cos (x))

Simplify

= - ln (1 + sin (x)) cos (x) + x + cos (x)

Add a constant to the solution

= - ln (1 + sin (x)) cos (x) + x + cos (x) + C

\int e^{- ik x} d x

\int \frac{1}{e ^{2 x} - 9} d x

\int \frac{1}{x} \frac{x + 1}{x} d x

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