What is the integral of (4/x+2e^x-sin(x)) ?

The integral of (4/x+2e^x-sin(x)) is 4ln|x|+2e^x+cos(x)+C

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integral of (4/x+2e^x-sin(x))

\int (\frac{4}{x} + 2 e^{x} - sin (x)) d x

4 ln ∣ x ∣ + 2 e^{x} + cos (x) + C

Solution steps

\int \frac{4}{x} + 2 e^{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 \frac{4}{x} d x + \int 2 e^{x} d x - \int sin (x) d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 2 e^{x} d x = 2 e^{x}

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

= 4 ln ∣ x ∣ + 2 e^{x} - (- cos (x))

Simplify

= 4 ln ∣ x ∣ + 2 e^{x} + cos (x)

Add a constant to the solution

= 4 ln ∣ x ∣ + 2 e^{x} + cos (x) + C

What is the integral of (4/x+2e^x-sin(x)) ?
The integral of (4/x+2e^x-sin(x)) is 4ln|x|+2e^x+cos(x)+C