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

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

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

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

x ln (x) - x - ln (x) cos (x) + Ci (x) + C

Solution steps

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

Expand

(1 + sin (x)) ln (x) : ln (x) + ln (x) sin (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int ln (x) d x = x ln (x) - x

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

= x ln (x) - x - ln (x) cos (x) + Ci (x)

Add a constant to the solution

= x ln (x) - x - ln (x) cos (x) + Ci (x) + C

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