What is the integral of ln(e^{(1-sin(x))/3}) ?

The integral of ln(e^{(1-sin(x))/3}) is xln(e^{1/3 (1-sin(x))})+1/3 (xsin(x)+cos(x))+C

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integral of ln(e^{(1-sin(x))/3})

\int ln (e^{\frac{1 - s i n ( x )}{3}}) d x

x ln (e^{\frac{1}{3} (1 - s i n (x))}) + \frac{1}{3} (x sin (x) + cos (x)) + C

Solution steps

\int ln (e^{\frac{1 - s i n ( x )}{3}}) d x

Apply Integration By Parts

: x ln (e^{\frac{1}{3} (1 - s i n (x))}) - \int - \frac{1}{3} x cos (x) d x

= x ln (e^{\frac{1}{3} (1 - s i n (x))}) - \int - \frac{1}{3} x cos (x) d x

\int - \frac{1}{3} x cos (x) d x = - \frac{1}{3} (x sin (x) + cos (x))

= x ln (e^{\frac{1}{3} (1 - s i n (x))}) - (- \frac{1}{3} (x sin (x) + cos (x)))

Simplify

= x ln (e^{\frac{1}{3} (1 - s i n (x))}) + \frac{1}{3} (x sin (x) + cos (x))

Add a constant to the solution

= x ln (e^{\frac{1}{3} (1 - s i n (x))}) + \frac{1}{3} (x sin (x) + cos (x)) + C

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What is the integral of ln(e^{(1-sin(x))/3}) ?
The integral of ln(e^{(1-sin(x))/3}) is xln(e^{1/3 (1-sin(x))})+1/3 (xsin(x)+cos(x))+C