What is the integral of 5/(yln(y))+xsin(y)+x ?

The integral of 5/(yln(y))+xsin(y)+x is 5ln|ln(y)|+xy-xcos(y)+C

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integral of 5/(yln(y))+xsin(y)+x

\int \frac{5}{y ln ( y )} + x sin (y) + x d y

5 ln ∣ ln (y) ∣ + x y - x cos (y) + C

Solution steps

\int \frac{5}{y ln ( y )} + x sin (y) + x d y

Apply the Sum Rule:

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

= \int \frac{5}{ln ( y ) y} d y + \int x d y + \int sin (y) x d y

\int \frac{5}{ln ( y ) y} d y = 5 ln ∣ ln (y) ∣

\int x d y = x y

\int sin (y) x d y = - x cos (y)

= 5 ln ∣ ln (y) ∣ + x y - x cos (y)

Add a constant to the solution

= 5 ln ∣ ln (y) ∣ + x y - x cos (y) + C

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What is the integral of 5/(yln(y))+xsin(y)+x ?
The integral of 5/(yln(y))+xsin(y)+x is 5ln|ln(y)|+xy-xcos(y)+C