What is the integral of ln(x)-cos(4x) ?

The integral of ln(x)-cos(4x) is xln(x)-x-1/4 sin(4x)+C

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integral of ln(x)-cos(4x)

\int ln (x) - cos (4 x) d x

x ln (x) - x - \frac{1}{4} sin (4 x) + C

Solution steps

\int ln (x) - cos (4 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 cos (4 x) d x

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

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= x ln (x) - x - \frac{1}{4} sin (4 x)

Add a constant to the solution

= x ln (x) - x - \frac{1}{4} sin (4 x) + C

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What is the integral of ln(x)-cos(4x) ?
The integral of ln(x)-cos(4x) is xln(x)-x-1/4 sin(4x)+C